03 Statistical Mechanics - conocimientos.com.ve: Betancourt Morales Bárbara Scarlett

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lunes, 15 de febrero de 2010

Introducción

Como muchas disciplinas, la termodinámica surge de los procedimientos empíricos que llevaron a la construcción de elementos que terminaron siendo muy útiles para el desarrollo de la vida del hombre.

Creemos que la termodinámica es un caso muy especial debido a que sus inicios se pierden en la noche de los tiempos mientras que en la actualidad los estudios sobre el perfeccionamiento de las máquinas térmicas siguen siendo de especial importancia, mas aun si tomamos en cuenta la importancia que revisten temas de tanta actualidad como la contaminación.

El origen fué sin lugar a dudas la curiosidad que despertara el movimiento producido por la energía del vapor de agua.

Su desarrollo fué tomando como objetivo principal el perfeccionamiento de las tecnologias aplicadas con el fin de hacer mas facil la vida del hombre, reemplazando el trabajo manual por la máquina que facilitaba su realización y lograba mayor rapidez, estos avances que gravitaban directamente en la economía, por ello el inicio se encuentra en el bombeo de aguas del interior de las minas y el transporte.

Mas tarde se intensificaron los esfuerzos por lograr el máximo de rendimiento lo que llevó a la necesidad de lograr un conocimiento profundo y acabado de las leyes y principios que regian las operaciones realizadas con el vapor.

El campo de la termodinámica y su fuente primitiva de recursos se amplía en la medida en que se incorporan nuevas áreas como las referentes a los motores de combustión interna y ultimamente los cohetes. La construcción de grandes calderas para producir enormes cantidades de trabajo marca tambien la actualidad de la importancia del binomio máquinas térmicas-termodinámica.

En resumen: en el comienzo se partió del uso de las propiedades del vapor para succionar agua de las minas, con rendimientos insignificantes, hoy se trata de lograr las máximas potencias con un mínimo de contaminación y un máximo de economía.

Para realizar una somera descripción del avance de la termodinámica a través de los tiempos la comenzamos identificando con las primitivas máquinas térmicas y dividimos su descripción en tres etapas, primero la que dimos en llamar empírica, la seguna la tecnológica y la tercera la científica.

I.- La etapa empírica

Los orígenes de la termodinámica nacen de la pura experiencia y de hallazgos casuales que fueron perfeccionándose con el paso del tiempo.

Algunas de las máquinas térmicas que se construyeron en la antigüedad fueron tomadas como mera curiosidad de laboratorio, otros se diseñaron con el fin de trabajar en propósitos eminentemente prácticos. En tiempos del del nacimiento de Cristo existian algunos modelos de máquinas térmicas, entendidas en esa época como instrumentos para la creación de movimientos autónomos, sin la participación de la tracción a sangre.

El ingenio más conocidos por las crónicas de la época es la eolipila de Herón que usaba la reacción producida por el vapor al salir por un orificio para lograr un movimiento. Esta máquina es la primera aplicacióndel principio que usan actualmente las llamadas turbinas de reacción.

La historia cuenta que en 1629 Giovanni Branca diseñó una máquina capaz de realizar un movimiento en base al impulso que producía sobre una rueda el vapor que salía por un caño. No se sabe a ciencia cierta si la máquina de Branca se construyó, pero, es claro que es el primer intento de construcción de las que hoy se llaman turbinas de acción.

La mayor aplicación de las posibilidades de la máquina como reemplazante de la tracción a sangre consistía en la elevación de agua desde el fondo de las minas. Por ello la primera aplicación del trabajo mediante la fuerza del vapor cristaliza en la llamada máquina de fuego de Savery.

La máquina de Savery consistía en un cilindro conectado mediante una cañería a la fuente de agua que se deseaba bombear, el cilindro se llenaba de vapor de agua, se cerraba la llave de ingreso y luego se enfriaba, cuando el vapor se condensaba se producía un vacío que permitía el ascenso del agua.

II.- La etapa tecnológica.

Según lo dicho la bomba de Savery no contenía elementos móviles, excepto las válvulas de accionamiento manual, funcionaba haciendo el vacío, de la misma manera en que ahora lo hacen las bombas aspirantes, por ello la altura de elevación del agua era muy poca ya que con un vacío perfecto se llegaría a lograr una columna de agua de 10.33 metros, pero, la tecnología de esa época no era adecuada para el logro de vacios elevados.

El primer aparato elemento que podriamos considerar como una máquina propiamente dicha, por poseer partes móviles, es la conocida como máquina de vapor de Thomas Newcomen construída en 1712. La innovación consistió en la utilización del vacío del cilindro para mover un pistón que a su vez proveía movimiento a un brazo de palanca que actuaba sobre una bomba convencional de las llamadas aspirante-impelente.

Podemos afirmar que es la primera máquna alternativa de mla que se tiene conocimiento y que con ella comienza la historia de las máquinas térmicas.

Las dimensiones del cilindro, órgano principal para la creación del movimien-to, eran: 53,3 cm de diámetro y 2,4 metros de altura, producía 12 carreras por minuto y elevaba 189 litros de agua desde una profundidad de 47,5 metros.

El principal progreso que se incorpora con la máquina de Newcomen consis-te en que la producción de un movimiento oscilatorio habilita el uso de la máquina para otros servicios que requieran movimiento alternativo, es decir, de vaivén.

En esa época no existian métodos que permitieran medir la potencia desarrollada por las máquinas ni unidades que permitieran la comparación de su rendi-miento, no obstante, los datos siguientes dan una idea del trabajo realizado por una máquina que funcionó en una mina en Francia, contaba con un cilindro de 76 cm de diámetro y 2,7 metros de altura, con ella se pudo completar en 48 horas una labor de desagote que previamente había requerido una semana con el traba-jo de 50 hombres y 20 caballos operando en turnos durante las 24 horas del día.

La máquina de Newcomen fué perfeccionada por un ingeniero inglés llamado Johon Smeaton (1742-1792). Un detalle de la potencia lograda lo podemos ver en el trabajo encargado por Catalina II de Rusia quien solicitó bombear agua a los di-ques secos del fuerte de Kronstadt. Esta tarea demoraba un año usando molinos de viento de 100 metros de altura, la máquina de Smeaton demoró solamente dos semanas. Se debe destacar que el perfeccionamiento consistió en la optimización de los mecanismos, cierres de válvulas, etc.

El análisis de las magnitudes que entran en juego en el funcionamiento de la máquina de vapor y su cuantificación fué introducido por James Watt (1736-1819).

Watt se propuso estudiar la magnitud del calor puesto en juego en el funcio-namiento de la máquina, esto permitiría estudiar su rendimiento.

El mayor obstáculo que encontró Watt fué el desconocimiento de los valores de las constantes físicas involucradas en el proceso, a raiz de ello debió realizar un proceso de mediciones para contar con datos confiables.

Sus mediciones experimentales le permitieron verificar que la máquina de Newcomen solo usaba un 33% del vapor consumido para realizar el trabajo útil.

Los aportes de Watt son muchos, todos ellos apuntaron al logro de un mayor rendimiento, inventó el prensaestopa que actua manteniendo la presión mientras se mueve el bástago del pistón, introdujo la bomba de vacío para incrementar el rendimiento en el escape, ensayó un mecanismo que convirtiera el movimiento alternativo en rotacional, en 1782 patentó la máquina de doble efecto (el vapor empuja en ambas carreras del pistón), ideó válvulas de movimiento vertical que permitian mantener la presión de la caldera mediante la fuerza de un resorte com-primido. Creó el manómetro para medir la presión del vapor y un indicador que po-día dibujar la evolución presión-volumen del vapor en el cilindro a lo largo de un ciclo.

Con el objetivo de establecer una unidad adecuada para la medición de la potencia, realizó experiencias para definir el llamado caballo de fuerza. Determinó que un caballo podía desarrollar una potencia equivalente a levantar 76 kg hasta una altura de 1 metro en un segundo, siguiendo con este ritmo durante cierto tiempo, este valor se usa actualmente y se lo llama caballo de fuerza inglés.

Un detalle importante de las calderas de Watt es que trabajaban a muy baja presión, 0,3 a 0,4 kg/cm2.

Los progresos tecnológicos aportados por Watt llevaron la tecnología de la máquina de vapor a un refinamiento considerable. Se había avanzado en seguri-dad merced a la incorporación de válvulas, ya se contaba con unidades que daban cuenta de la potencia y el rendimiento, los mecanismos fueron elaborados con los mas recientes avances de la tecnología mecánica. Lo único que no entró en la consideración de Watt fué la posibilidad de usar calderas de mayor presión, su objetivo principal era la seguridad, y desde el punto de vista económico no reque-ría perfeccionamiento, sus máquinas eran muy apreciadas y se vendian bien.

Después de Watt se consiguieron considerables avances en la utilización de calderas de muy alta presión, esta incorporación incrementó el rendimiento y, lo mas importante, favoreció el uso de calderas de menor tamaño que realizaban mayor trabajo que las grandes, además de mejorar el rendimiento del vapor las preparó para adaptarlas para su instalación en medios de transporte.

En agosto de 1807 Robert Fulton puso en funcionamiento el primer barco de vapor de éxito comercial, el Clermont, el mérito de Fulton consiste en la instalación y puesta en marcha de una máquina de vapor a bordo, no realizó innovaciones sobre la máquina en sí. Este barco cumplió un servicio fluvial navegando en el río Hudson.

En el año 1819 el buque de vapor Savannah, de bandera norteamericana realiza el primer viaje transatlántico, ayudado por un velamen. El Britania fué el primer barco de vapor inglés, entró en servicio en 1840, desplazaba 1150 toneladas y contaba con una máquina de 740 caballos de fuerza, alimentada por cuatro calderas de 0.6 kg/cm cuadrado, desarrollando una velocidad de 14 km/h.

George Stephenson (1781-1848) fué el primero que logró instalar una máquina de vapor sobre un vehículo terrestre dando inicio a la era del ferrocarril.

En el año 1814 Stephenson logró arrastrar una carga de treinta toneladas por una pendiente de 1 en 450 a sis km por hora.
En 1829 la locomotora llamada Rocket recorrió 19 km en 53 minutos lo que fué un record para la época.

III.- Etapa científica.

Sadi Carnot (1796-1832) es el fundador de la termodinámica como disciplina teórica, escribió su trabajo cumbre a los 23 años. Este escrito estuvo desconocido durante 25 años hasta que el físico Lord Kelvin redescubriera la importancia de las propuestas contenidas en él.

Llamó la atención de Carnot el hecho de que no existieran teorias que ava-laran la propuestas utilizadas en el diseño de las máquinas de vapor y que todo ello dependira de procedimientos enteramente empíricos. Para resolver la cuestión propuso que se estudiara todo el procedimiento desde el punto de vista mas gene-ral, sin hacer referencia a un motor, máquina o fluido en especial.

Las bases de las propuestas de Carnot se pueden resumir haciendo notar que fué quien desarrolló el concepto de proceso cíclico y que el trabajo se produ-cía enteramente "dejando caer" calor desde una fuente de alta temperatura hasta un depósito a baja temperatura. También introdujo el concepto de máquina reversible.

El principio de Carnot establece que la máxima cantidad de trabajo que puede ser producido por una máquina térmica que trabaja entre una fuente a alta temperatura y un depósito a temperatura menor, es el trabajo producido por una máquina reversible que opere entre esas dos temperaturas. Por ello demostró que ninguna máquina podía ser mas eficiente que una máquina reversible.

A pesar que estas ideas fueron expresadas tomando como base la teoría del calórico, resultaron válidas. Posteriormente Clausius y Kelvin, fundadores de la termodinámica teórica, ubicaron el principio de Carnot dentro de una rigurosa teo-ría científica estableciendo un nuevo concepto, el segundo principio de la termodinámica.

Carnot también establece que el rendimiento de cualquier máquina térmica depende de la diferencia entre temperatura de la fuente mas caliente y la fría. Las altas temperaturas del vapor presuponen muy altas presiones y la expansión del vapor a bajas temperaturas producen grandes volúmenes de expansión. Esto producía una cota en el rendimiento y la posibilidad de construcción de máquinas de vapor.

En esta época todavía tenía vigencia la teoría del calórico, no obstante ya estaba germinando la idea de que esa hipótesis no era la adecuada, en el marco de las sociedades científicas las discusiones eran acaloradas.

James Prescot Joule (1818-1889) se convenció rapidamente de que el trabajo y el calor eran diferentes manifestaciones de una misma cosa. Su expe-riencia mas recordada es aquella en que logra medir la equivalencia entre el traba-jo mecánico y la cantidad de calor. Joule se valió para esta experiencia de un sis-tema de hélices que agitaban el agua por un movimiento producido por una serie de contrapesos que permitian medir la energía mecánica puesta en juego.

A partir de las investigaciones de Joule se comenzó a debilitar la teoría del calórico, en especial en base a los trabajos de Lord Kelvin quien junto a Clausius terminaron de establecer las bases teóricas de la termodinámica como disciplina independiente. En el año 1850 Clausius dscubrió la existencia de la entropía y enunció el segundo principio:

Es imposible que una máquina térmica que actúa por sí sola sin recibir ayuda de ningún agente externo, transporte calor de un cuerpo a otro que está a mayor temperatura.

En 1851 Lord Kelvin publicó un trabajo en el que compatibilizaba los estudios de Carnot, basados en el calórico, con las conclusiones de Joule, el calor es una forma de energía, compartió las investigaciones de Clausius y reclamó para sí el postulado del primer principio que enunciaba así:

Es imposible obtener, por medio de agentes materiales inanimados, efectos mecánicos de cualquier porción de materia enfriándola a una temperatura inferior a la de los objetos que la rodean.

Lord Kelvin también estableció un principio que actualmente se conoce como el primer principio de la termodinámica. Y junto a Clausius derrotaron la teoría del calórico.

Situación Actual

Hoy se ha llegado a uninteresante perfeccionamiento de las máquinas térmicas, sobre una teoría basada en las investigaciones de Clausius, Kelvin y Carnot, cuyos principios están todavía en vigencia, la variedad de máquinas térmicas va desde las grandes calderas de las centrales nucleares hasta los motores cohete que impulsan los satélites artificiales, pasando por el motor de explosión, las turbinas de gas, las turbinas de vapor y los motores de retropropulsión. Por otra parte la termodinámica como ciencia actua dentro de otras disciplinas como la química, la biología, etc.

Conclusión

El desarrollo de la termodinámica tiene un origen empírico como muchas de las partes de la tecnología.

Una de las curiosidades en la aplicación temprana de efectos del vapor en la etapa que dimos en llamar empírica y que a lo largo de su desarrollo cambiara su origen en varias hipótesis, flogisto, calórico y finalmente energía.

Con Watt se logra el perfeccionamiento en la tecnología, se comprenden los principios básicos de la misma y se aislan las variables que intervienen en el fun-cionamiento de la máquina, la introducción de la unidad para medir la potencia conduce al manejo de criterios de comparación.

Despues de Watt comienza el desarrollo de las máquinas móviles con las realizaciones de Robert Fulton y George Stephenson.

Tambien es importante marcar como las teorias de Carnot tienen aún validez en su forma original apesar de haber estado fundamentadas en una hipótesis erro-nea, la del calórico. Carnot introduce tres conceptos fundamentales:

El concepto de ciclo o máquina cíclica.

La relación entre la "caida del calor de una fuente caliente a otra mas fría y su relación con el trabajo.
El concepto de máquina reversible de rendimiento máximo.
Gracias a Clausius y Kelvin se convierte a la termodinámica en una ciencia independiente de alto contenido teórico y matemático, lo que logra entender los fenómenos que se desarrollaban y fundamentar progresos tecnológicos.

Bibliografía de referencia

Motores térmicos e hidráulicos	Rosich	Ergon
Termodinámica Técnica	Estrada	Editorial Alsina.
Maquinas Térmicas	Sandfort	Eudeba
A TextBook on Heat	Barton	Longsman
Heat	Mitton	Dent and sons

Las palabras clave se refieren a los precursores de esta ciencia: Herón, Savery, Newcomen, Fulton, Stephenson, Sadi Carnot, Clausius, Lord Kelvin, Joule, Watt.

Bárbara Scarlett Betancourt Morales

sábado, 13 de febrero de 2010

Primera ley de la Termodinámica

La primera ley no es otra cosa que el principio de conservación de la energía aplicado a un sistema de muchísimas partículas. A cada estado del sistema le corresponde una energía interna U. Cuando el sistema pasa del estado A al estado B, su energía interna cambia en

DU=U_B-U_A Supongamos que el sistema está en el estado A y realiza un trabajo W, expandiéndose. Dicho trabajo mecánico da lugar a un cambio (disminución) de la energía interna de sistema
DU=-W
También podemos cambiar el estado del sistema poniéndolo en contacto térmico con otro sistema a diferente temperatura. Si fluye una cantidad de calor Q del segundo al primero, aumenta su energía interna en
DU=Q
Si el sistema experimenta una transformación cíclica, el cambio en la energía interna es cero, ya que se parte del estado A y se regresa al mismo estado, DU=0. Sin embargo, durante el ciclo el sistema ha efectuado un trabajo, que ha de ser proporcionado por los alrededores en forma de transferencia de calor, para preservar el principio de conservación de la energía, W=Q.

Si la transformación no es cíclica DU¹ 0
Si no se realiza trabajo mecánico DU=Q
Si el sistema está aislado térmicamente DU=-W
Si el sistema realiza trabajo, U disminuye
Si se realiza trabajo sobre el sistema, U aumenta
Si el sistema absorbe calor al ponerlo en contacto térmico con un foco a temperatura superior, U aumenta.
Si el sistema cede calor al ponerlo en contacto térmico con un foco a una temperatura inferior, U disminuye.

Todo estos casos, los podemos resumir en una única ecuación que describe la conservación de la energía del sistema.

DU=Q-W

Si el estado inicial y final están muy próximos entre sí, el primer principio se escribe dU=dQ-pdV

Transformaciones

La energía interna U del sistema depende únicamente del estado del sistema, en un gas ideal depende solamente de su temperatura. Mientras que la transferencia de calor o el trabajo mecánico dependen del tipo de transformación o camino seguido para ir del estado inicial al final.

Isócora o a volumen constante

No hay variación de volumen del gas, luego W=0
Q=nc_V(T_B-T_A)
Donde c_V es el calor específico a volumen constante

Isóbara o a presión constante

W=p(v_B-v_A) Q=nc_P(T_B-T_A)
Donde c_P es el calor específico a presión constante

Calores específicos a presión constante c_P y a volumen constante c_V

En una transformación a volumen constante dU=dQ=nc_VdT En una transformación a presión constante dU=nc_PdT-pdV Como la variación de energía interna dU no depende del tipo de transformación, sino solamente del estado inicial y del estado final, la segunda ecuación se puede escribir como nc_VdT=nc_PdT-pdV Empleando la ecuación de estado de un gas ideal pV=nRT, obtenemos la relación entre los calores específicos a presión constante y a volumen constante c_V=c_P-R Para un gas monoatómico

Para un gas diatómico

La variación de energía interna en un proceso AB es DU=nc_V(T_B-T_A) Se denomina índice adiabático de un gas ideal al cociente

Isoterma o a temperatura constante

pV=nRT La curva p=cte/V que representa la transformación en un diagrama p-Ves una hipérbola cuyas asíntotas son los ejes coordenados.

DU=0
Q=W

Adiabática o aislada térmicamente, Q=0

La ecuación de una transformación adiabática la hemos obtenido a partir de un modelo simple de gas ideal. Ahora vamos a obtenerla a partir del primer principio de la Termodinámica.

Ecuación de la transformación adiabática Del primer principio dU=-pdV

Integrando

Donde el exponente de V se denomina índice adiabático g del gas ideal

Si A y B son los estados inicial y final de una transformación adiabática se cumple que

Para calcular el trabajo es necesario efectuar una integración similar a la transformación isoterma.

Como podemos comprobar, el trabajo es igual a la variación de energía interna cambiada de signo Si Q=0, entonces W=-DU=-nc_V(T_B-T_A)

Barbara Betancourt

viernes, 12 de febrero de 2010

The second law of Thermodynamics

The second law of thermodynamics is an expression of the universal principle of entropy, stating that the entropy of an isolated system which is not in equilibrium will tend to increase over time, approaching a maximum value at equilibrium; and that the entropy change dS of a system undergoing any infinitesimal reversible process is given by δq / T, where δq is the heat supplied to the system and T is the absolute temperature of the system. In classical thermodynamics, the second law is taken to be a basic postulate, while in statistical thermodynamics, the second law is a consequence of applying the fundamental postulate, also known as the equal a priori probability postulate,^{[clarification needed]} to the future while empirically accepting that the past was low entropy, for reasons not yet well understood.^{[clarification needed]}

The origin of the second law can be traced to French physicist Sadi Carnot's 1824 paper Reflections on the Motive Power of Fire, which presented the view that motive power (work) is due to the flow of caloric (heat) from a hot to cold body (working substance). In simple terms, the second law is an expression of the fact that over time, ignoring the effects of self-gravity, differences in temperature, pressure, and chemical potential tend to even out in a physical system that is isolated from the outside world. Entropy is a measure of how much this evening-out process has progressed.

There are many versions of the second law, but they all have the same effect, which is to explain the phenomenon of irreversibility in nature.

Versions of the Law

There are many ways of stating the second law of thermodynamics, but all are equivalent in the sense that each form of the second law logically implies every other form.Thus, the theorems of thermodynamics can be proved using any form of the second law and third law.
The formulation of the second law that refers to entropy directly is as follows:

In a system, a process that occurs will tend to increase the total entropy of the universe.

Thus, while a system can go through some physical process that decreases its own entropy, the entropy of the universe (which includes the system and its surroundings) must increase overall. (An exception to this rule is a reversible or "isentropic" process, such as frictionless adiabatic compression.) Processes that decrease the total entropy of the universe are impossible. If a system is at equilibrium, by definition no spontaneous processes occur, and therefore the system is at maximum entropy.
A second formulation, due to Rudolf Clausius, is the simplest formulation of the second law, the heat formulation or Clausius statement:

Heat generally cannot flow spontaneously from a material at lower temperature to a material at higher temperature.

Informally, "Heat doesn't flow from cold to hot (without work input)", which is true obviously from ordinary experience. For example in a refrigerator, heat flows from cold to hot, but only when aided by an external agent (i.e. the compressor). Note that from the mathematical definition of entropy, a process in which heat flows from cold to hot has decreasing entropy. This can happen in a non-isolated system if entropy is created elsewhere, such that the total entropy is constant or increasing, as required by the second law. For example, the electrical energy going into a refrigerator is converted to heat and goes out the back, representing a net increase in entropy.
The exception to this is for statistically unlikely events where hot particles will "steal" the energy of cold particles enough that the cold side gets colder and the hot side gets hotter, for an instant. Such events have been observed at a small enough scale where the likelihood of such a thing happening is significant.^[2] The mathematics involved in such an event are described by fluctuation theorem.
A third formulation of the second law, by Lord Kelvin, is the heat engine formulation, or Kelvin statement:

It is impossible to convert heat completely into work in a cyclic process.

That is, it is impossible to extract energy by heat from a high-temperature energy source and then convert all of the energy into work. At least some of the energy must be passed on to heat a low-temperature energy sink. Thus, a heat engine with 100% efficiency is thermodynamically impossible.
A fourth version of the second law was deduced by the Greek mathematician Constantin Carathéodory. The Carathéodory statement:

In the neighbourhood of any equilibrium state of a thermodynamic system, there are equilibrium states that are adiabatically inaccessible.

A final version of the second law was put to rhyme by Flanders and Swann^[3], based on the Clausius statement:

Heat won't pass from a cooler to a hotter
You can try it if you like but you far better notter
'cos the cold in the cooler will get hotter as a ruler
'cos the hotter body's heat will pass to the cooler!

Formulations of the second law in modern textbooks that introduce entropy from the statistical point of view, often contain two parts. The first part states that the entropy of an isolated system cannot decrease, or, to be more precise, the probability of an entropy increase is exceedingly small. The second part gives the relation between infinitesimal entropy increase of a system and an infinitesimal amount of absorbed heat in case of an arbitrary infinitesimal reversible process:

d S = δ q / T

. The reason why these two statements are not combined into a single statement is because the first part refers to a general non-equilibrium process in which temperature is not defined.

Microscopic systems

Thermodynamics is a theory of macroscopic systems and therefore the second law applies only to macroscopic systems with well-defined temperatures. For example, in a system of two molecules, there is a non-trivial probability that the slower-moving ("cold") molecule transfers energy to the faster-moving ("hot") molecule. Such tiny systems are not part of classical thermodynamics, but they can be investigated by quantum thermodynamics by using statistical mechanics. For any isolated system with a mass of more than a few picograms, probabilities of observing a decrease in entropy approach zero.

Energy dispersal

The second law of thermodynamics is an axiom of thermodynamics concerning heat, entropy, and the direction in which thermodynamic processes can occur. For example, the second law implies that heat does not flow spontaneously from a cold material to a hot material, but it allows heat to flow from a hot material to a cold material. Roughly speaking, the second law says that in an isolated system, concentrated energy disperses over time, and consequently less concentrated energy is available to do useful work. Energy dispersal also means that differences in temperature, pressure, and density even out. Again roughly speaking, thermodynamic entropy is a measure of energy dispersal, and so the second law is closely connected with the concept of entropy.

Overview

In a general sense, the second law is that temperature differences between systems in contact with each other tend to equalize and that work can be obtained from these non-equilibrium differences, but that loss of heat occurs, in the form of entropy, when work is done.^[5] Pressure differences, density differences, and particularly temperature differences, all tend to equalize if given the opportunity. This means that an isolated system will eventually have a uniform temperature. A heat engine is a mechanical device that provides useful work from the difference in temperature of two bodies:

Heat engine diagram

During the 19th century, the second law was synthesized, essentially, by studying the dynamics of the Carnot heat engine in coordination with James Joule's mechanical equivalent of heat experiments. Since any thermodynamic engine requires such a temperature difference, it follows that useful work cannot be derived from an isolated system in equilibrium; there must always be an external energy source and a cold sink. By definition, perpetual motion machines of the second kind would have to violate the second law to function.

History

Informal descriptions

The second law can be stated in various succinct ways, including:

It is impossible to produce work in the surroundings using a cyclic process connected to a single heat reservoir (Kelvin, 1851).
It is impossible to carry out a cyclic process using an engine connected to two heat reservoirs that will have as its only effect the transfer of a quantity of heat from the low-temperature reservoir to the high-temperature reservoir (Clausius, 1854).
If thermodynamic work is to be done at a finite rate, free energy must be expended.

Mathematical descriptions

In 1856, the German physicist Rudolf Clausius stated what he called the "second fundamental theorem in the mechanical theory of heat" in the following form:^[10]

$\int \frac{\delta Q}{T} = -N$

where Q is heat, T is temperature and N is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:

The entropy of the universe tends to a maximum.

This statement is the best-known phrasing of the second law. Moreover, owing to the general broadness of the terminology used here, e.g. universe, as well as lack of specific conditions, e.g. open, closed, or isolated, to which this statement applies, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This, of course, is not true; this statement is only a simplified version of a more complex description.
In terms of time variation, the mathematical statement of the second law for an isolated system undergoing an arbitrary transformation is:

$\frac{dS}{dt} \ge 0$

where

S is the entropy and

t is time.

It should be noted that statistical mechanics gives an explanation for the second law by postulating that a material is composed of atoms and molecules which are in constant motion. A particular set of positions and velocities for each particle in the system is called a microstate of the system and because of the constant motion, the system is constantly changing its microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is equally likely to occur, and when this assumption is made, it leads directly to the conclusion that the second law must hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the order of 1/√N where N is the number of particles in the system. For everyday (macroscopic) situations, the probability that the second law will be violated is practically zero. However, for systems with a small number of particles, thermodynamic parameters, including the entropy, may show significant statistical deviations from that predicted by the second law. Classical thermodynamic theory does not deal with these statistical variations.

Available useful work

An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an unlimited heat reservoir at temperature T_R and pressure P_R — so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain T_R; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain P_R.
Whatever changes to dS and dS_R occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy S_tot of the isolated total system must not decrease:

$dS_{\mathrm{tot}}= dS + dS_R \ge 0$

According to the First Law of Thermodynamics, the change dU in the internal energy of the sub-system is the sum of the heat δq added to the sub-system, less any work δw done by the sub-system, plus any net chemical energy entering the sub-system d ∑μ_iRN_i, so that:

$dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,$

where μ_iR are the chemical potentials of chemical species in the external surroundings.
Now the heat leaving the reservoir and entering the sub-system is

$\delta q = T_R (-dS_R) \le T_R dS$

where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the Second Law inequality from above.
It therefore follows that any net work δw done by the sub-system must obey

$\delta w \le - dU + T_R dS + \sum \mu_{iR} dN_i \,$

It is useful to separate the work δw done by the subsystem into the useful work δw_u that can be done by the sub-system, over and beyond the work p_R dV done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work that can be done:

$\delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\,$

It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the availability or exergy X of the subsystem,

$X = U - T_R S + p_R V - \sum \mu_{iR} N_i$

The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,

$d X + \delta w_u \le 0 \,$

i.e. the change in the subsystem's exergy plus the useful work done by the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done on the system) must be less than or equal to zero.

Special cases: Gibbs and Helmholtz free energies

When no useful work is being extracted from the sub-system, it follows that

$d X \le 0 \,$

with the exergy X reaching a minimum at equilibrium, when dX=0.
If no chemical species can enter or leave the sub-system, then the term ∑ μ_iR N_i can be ignored. If furthermore the temperature of the sub-system is such that T is always equal to T_R, then this gives:

$X = U - TS + p_R V + \mathrm{const.} \,$

If the volume V is constrained to be constant, then

$X = U - TS + \mathrm{const.'} = A + \mathrm{const.'}\,$

where A is the thermodynamic potential called Helmholtz free energy, A=U−TS. Under constant-volume conditions therefore, dA ≤ 0 if a process is to go forward; and dA=0 is the condition for equilibrium.
Alternatively, if the sub-system pressure p is constrained to be equal to the external reservoir pressure p_R, then

$X = U - TS + pV + \mathrm{const.} = G + \mathrm{const.}\,$

where G is the Gibbs free energy, G=U−TS+PV. Therefore under constant-pressure conditions, if dG ≤ 0, then the process can occur spontaneously, because the change in system energy exceeds the energy lost to entropy. dG=0 is the condition for equilibrium. This is also commonly written in terms of enthalpy, where H=U+PV.

Application

In sum, if a proper infinite-reservoir-like reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in X for an irreversible process and no change for a reversible process.

$dS_{tot} \ge 0$ is equivalent to $dX + \delta W_u \le 0$

This expression together with the associated reference state permits a design engineer working at the macroscopic scale (above the thermodynamic limit) to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. (Also, see process engineer). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found (See second law efficiency.)
This approach to the Second Law is widely utilized in engineering practice, environmental accounting, systems ecology, and other disciplines.

Proof of the Second Law

As mentioned above, in statistical mechanics, the Second Law is not a postulate, rather it is a consequence of the fundamental postulate, also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy. The first part of the second law, which states that the entropy of a thermally isolated system can only increase is a trivial of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy of

E

is:

$S = k \log\left[\Omega\left(E\right)\right]\,$

where $\Omega\left(E\right)$ is the number of quantum states in a small interval between

E

and

E + δ E

. Here

δ E

is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of

δ E

. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on

δ E

.
Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variable can e.g. refer to the total volume, the positions of pistons in the system etc.. Then

Ω

will depend on the values of these variables. If a variable is not fixed, e.g. we do not clamp a piston in a certain position, then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such that

Ω

is maximized as that is the most probable situation in equilibrium.
If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so that

Ω

is maximized, implies that that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value).
The entropy of a system that is not in equilibrium can be defined as:

$S = -k_{B}\sum_{j}P_{j}\log\left(P_{j}\right)$

see here. Here the

P j

are the probabilities for the system to be found in the states labeled by the subscript j. In thermal equilibrium the probabilities for states inside the energy interval

δ E

are all equal to

1 / Ω

, and in that case the general definition coincides with the previous definition of S that applies to the case of thermal equilibrium.
Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number

Ω

of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of

1 / Ω

. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's H-theorem, however, proves that the entropy will increase continuously as a function of time during the intermediate out of equilibrium state.

Proof of $dS =\frac{\delta Q}{T}$ for reversible processes

The second part of the Second Law states that the entropy change of a system undergoing a reversible processes is given by:

$dS =\frac{\delta Q}{T}$

where the temperature is defined as:
$\frac{1}{k T}\equiv\beta\equiv\frac{d\log\left[\Omega\left(E\right)\right]}{dE}\,$
See here for the justification for this definition. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the adiabatic theorem of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.
The generalized force, X, corresponding to the external variable x is defined such that

X d x

is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate

E r

is given by:

$X = -\frac{dE_{r}}{dx}$

Since the system can be in any energy eigenstate within an interval of

δ E

, we define the generalized force for the system as the expectation value of the above expression:

$X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,$

To evaluate the average, we partition the $\Omega\left(E\right)$ energy eigenstates by counting how many of them have a value for $\frac{dE_{r}}{dx}$ within a range between

Y

and

Y + δ Y

. Calling this number $\Omega_{Y}\left(E\right)$ , we have:

$\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,$

The average defining the generalized force can now be written:

$X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,$

We can relate this to the derivative of the entropy w.r.t. x at constant energy E as follows. Suppose we change x to x + dx. Then $\Omega\left(E\right)$ will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between

E

and

E + δ E

. Let's focus again on the energy eigenstates for which $\frac{dE_{r}}{dx}$ lies within the range between

Y

and

Y + δ Y

. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E - Y dx to E move from below E to above E. There are

$N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,$

such energy eigenstates. If $Y dx\leq\delta E$ , all these energy eigenstates will move into the range between

E

and

E + δ E

and contribute to an increase in

Ω

. The number of energy eigenstates that move from below

E + δ E

to above

E + δ E

is, of course, given by $N_{Y}\left(E+\delta E\right)$ . The difference

$N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,$

is thus the net contribution to the increase in

Ω

. Note that if Y dx is larger than

δ E

there will be the energy eigenstates that move from below E to above

E + δ E

. They are counted in both $N_{Y}\left(E\right)$ and $N_{Y}\left(E+\delta E\right)$ , therefore the above expression is also valid in that case.
Expressing the above expression as a derivative w.r.t. E and summing over Y yields the expression:

$\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,$

The logarithmic derivative of

Ω

w.r.t. x is thus given by:

$\left(\frac{\partial\log\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,$

The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:

$\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,$

Combining this with

$\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,$

Gives:

$dS = \left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{\partial x}\right)_{E}dx = \frac{dE}{T} + \frac{X}{T} dx=\frac{\delta Q}{T}\,$

Proof for systems described by the canonical ensemble

If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the canonical ensemble:

$P_{j}=\frac{\exp\left(-\frac{E_{j}}{k T}\right)}{Z}$

Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the partition function. We now consider an infintesimal reversible change in the temperature and in the external parameters on which the energy levels depend. It follows from the general formula for the entropy:

$S = -k\sum_{j}P_{j}\log\left(P_{j}\right)$

that

$dS = -k\sum_{j}\log\left(P_{j}\right)dP_{j}$

Inserting the formula for

P j

for the canonical ensemble in here gives:

$dS = \frac{1}{T}\sum_{j}E_{j}dP_{j}=\frac{1}{T}\sum_{j}d\left(E_{j}P_{j}\right) - \frac{1}{T}\sum_{j}P_{j}dE_{j}= \frac{dE + \delta W}{T}=\frac{\delta Q}{T}$

Criticisms

Owing to the somewhat ambiguous nature of the formulation of the second law, i.e. the postulate that the quantity heat divided by temperature increases in spontaneous natural processes, it has occasionally been subject to criticism as well as attempts to dispute or disprove it. Clausius himself even noted the abstract nature of the second law. In his 1862 memoir, for example, after mathematically stating the second law by saying that integral of the differential of a quantity of heat divided by temperature must be less than or equal to zero for every cyclical process which is in any way possible (Clausius Inequality)

$\oint \frac{\delta Q}{T} \le 0$ ,

Clausius then stated:

Although the necessity of this theorem admits of strict mathematical proof if we start from the fundamental proposition above quoted it thereby nevertheless retains an abstract form, in which it is with difficulty embraced by the mind, and we feel compelled to seek for the precise physical cause, of which this theorem is a consequence.

Recall that heat and temperature are statistical, macroscopic quantities that become somewhat ambiguous when dealing with a small number of atoms.

Perpetual motion of the second kind

Before 1850, heat was regarded as an indestructible particle of matter. This was called the "material hypothesis", as based principally on the purported^[11]^[12]^[13]^[14] views of Sir Isaac Newton. It was on these views, partially, that in 1824 Sadi Carnot formulated the initial version of the second law. It soon was realized, however, that if the heat particle was conserved, and as such not changed in the cycle of an engine, that it would be possible to send the heat particle cyclically through the working fluid of the engine and use it to push the piston and then return the particle, unchanged, to its original state. In this manner perpetual motion could be created and used as an unlimited energy source. Thus, historically, people have always been attempting to create a perpetual motion machine, in violation of the second law, in the hope of solving the world's energy limitations.

Maxwell's Demon

Main article: Maxwell's demon

In 1871, James Clerk Maxwell proposed a thought experiment, now called Maxwell's demon, which challenged the second law. This experiment reveals that the information theoretical definition of entropy is more fundamental than the definition according to classical thermodynamics.

Time's arrow and the origin of the second law

Main article: Entropy (arrow of time)

The second law is a law about macroscopic irreversibility, and so may appear to violate the principle of T-symmetry. Boltzmann first investigated the link with microscopic reversibility. In his H-theorem he gave an explanation, by means of statistical mechanics, for dilute gases in the zero density limit where the ideal gas equation of state holds. He derived the second law of thermodynamics not from mechanics alone, but also from the probability arguments. His idea was to write an equation of motion for the probability that a single particle has a particular position and momentum at a particular time. One of the terms in this equation accounts for how the single particle distribution changes through collisions of pairs of particles. This rate depends on the probability of pairs of particles. Boltzmann introduced the assumption of molecular chaos to reduce this pair probability to a product of single particle probabilities. From the resulting Boltzmann equation he derived his famous H-theorem, which implies that on average the entropy of an ideal gas can only increase.
The assumption of molecular chaos in fact violates time reversal symmetry. It assumes that particle momenta are uncorrelated before collisions. If you replace this assumption with "anti-molecular chaos," namely that particle momenta are uncorrelated after collision, then you can derive an anti-Boltzmann equation and an anti-H-Theorem which implies entropy decreases on average. Thus we see that in reality Boltzmann did not succeed in solving Loschmidt's paradox. The molecular chaos assumption is the key element that introduces the arrow of time.
The origin of the arrow of time is today usually thought to be the smooth, uncorrelated (and hence low entropy) initial conditions that existed in the very early universe.^[15]^[16]^[17]^[18]

Applications

Main article: Entropy and life

The second law of thermodynamics has been proven mathematically for thermodynamic systems, where entropy is defined in terms of heat divided by the absolute temperature. The second law is often applied to other situations, such as black hole thermodynamics , and the complexity of life, or orderliness. ^[19] However it is incorrect to apply the closed-system expression of the second law of thermodynamics to any one sub-system connected by mass-energy flows to another ("open system"). In sciences such as biology and biochemistry the application of thermodynamics is well-established, e.g. biological thermodynamics. The general viewpoint on this subject is summarized well by biological thermodynamicist Donald Haynie; as he states: "Any theory claiming to describe how organisms originate and continue to exist by natural causes must be compatible with the first and second laws of thermodynamics."^[20]
This is very different, however, from the claim made by many creationists that evolution violates the second law of thermodynamics. Evidence indicates that biological systems and evolution of those systems conform to the second law, since although biological systems may become more ordered, the net change in entropy for the entire universe is still positive as a result of evolution.^[21] Additionally, the process of natural selection responsible for such local increase in order may be mathematically derived from the expression of the second law equation for non-equilibrium connected open systems,^[22] arguably making the Theory of Evolution itself an expression of the Second Law.
Furthermore, the second law is only true of closed systems. It is easy to decrease entropy, with an energy source (such as the sun). For example, a refrigerator separates warm and cold air, but only when it is plugged in. Since all biology requires an external energy source, there's nothing unusual (thermodynamically) with it growing more complex with time.

Complex systems

It is occasionally claimed that the second law is incompatible with autonomous self-organisation, or even the coming into existence of complex systems. This is a common creationist argument against evolution.^[23] The entry self-organisation explains how this claim is a misconception. In fact, as hot systems cool down in accordance with the second law, it is not unusual for them to undergo spontaneous symmetry breaking, i.e. for structure to spontaneously appear as the temperature drops below a critical threshold. Complex structures, such as Bénard cells, also spontaneously appear where there is a steady flow of energy from a high temperature input source to a low temperature external sink.
Furthermore, a system that energy flows into and out of may decrease its local entropy provided the increase of the entropy to its surrounding that this process causes is greater than or equal to the local decrease in entropy. A good example of this is crystallization. As a liquid cools, crystals begin to form inside it. While these crystals are more ordered than the liquid they originated from, in order for them to form they must release a great deal of heat, known as the latent heat of fusion. This heat flows out of the system and increases the entropy of its surroundings to a greater extent than the decrease of energy that the liquid undergoes in the formation of crystals.
An interesting situation to consider is that of a supercooled liquid perfectly isolated thermodynamically, into which a grain of dust is dropped. Here even though the system cannot export energy to its surroundings, it will still crystallize. Now however the release of latent heat will contribute to raising its own temperature. If this release of heat causes the temperature to reach the melting point before it has fully crystallized, then it shall remain a mixture of liquid and solid; if not, then it will be a solid at a significantly higher temperature than it previously was as a liquid. In both cases entropy from its disordered structure is converted into entropy of disordered motion.^{[citation needed]}

Quotes

"The law that entropy always increases holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations — then so much the worse for Maxwell's equations. If it is found to be contradicted by observation — well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation." — Sir Arthur Stanley Eddington, The Nature of the Physical World (1927)

The tendency for entropy to increase in isolated systems is expressed in the second law of thermodynamics — perhaps the most pessimistic and amoral formulation in all human thought. — Gregory Hill and Kerry Thornley, Principia Discordia (1965)

There are almost as many formulations of the second law as there have been discussions of it. — Philosopher / Physicist P.W. Bridgman, (1941)

Barbara Scarlett Betancourt Morales

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