16.4 Thermodynamics

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Thermodynamics deals with the overall properties of systems such as a vessel of gas or mixture of gases, a block of ice, a magnetized iron bar, etc. While such systems may be impossibly complex at the microscopic level, their thermodynamic behaviour is governed by a very few number of variables. For example, the state of a simple gas is determined by two variables, its volume $V$ and pressure $p,$ , while a mixture of gases also requires specification of the molar concentrations $n_1,n_2,\dots$ . representing the relative number ofparticles ofeach species of gas. An iron bar may need information from among variables such as its length $\ell ,$ cross-section $A,$ tensile strength $f$ and Young’s modulus $Y_{:}$ , the magnetic field $H$, magne tization ${\mu }_{:}$ , electric field $E$ and conductivity $\sigma$ . In any case, the number of variables needed for a thermodynamic description of the system is tiny compared to the $10^{24}$ or so variables required for a complete description of the microscopic state of the system (see Section 14.4).

The following treatment is similar to that given in [10]. Every thermodynamic system will be assumed to have a special class of states known as equilibrium states, forming an $n$-dimensional manifold $K$ , and given locally by a set of thermodynamic variables $\mathbf{x}=(x^1,x^2,\dots ,x^n)$ . The dimension $n$ is called the number of degrees of freedom of the thermodynamic system. Physically, we think of an equilibrium state as one in which the system remains when all external forces are removed. For a perfect gas there are two degrees of freedom, usually set to be $x^1=p$ and $x^2=V$ . The variable $p$ is called an internal or thermal variable, characterized physically by the fact that no work is done on or by the system if we change $p$ alone, leaving $V$ unaltered. Variables such as volume $V_{;}$ , a change in which results in work being done on the system, are called external or deformation variables.

A quasi-static or reversible process, resulting in a transition from one equilibrium state ${\mathbf{X}}_1$ to another ${\mathbf{X}}_2.$ , is a parametrized curve $\gamma :[t_1,t_2]K$ such that $\gamma (t_1)={\mathbf{x}}_1$ and $\gamma (t_2)={\mathbf{x}}_2$ . Since the curve passes through a continuous succession of equilibrium states, it should be thought ofas occurring infinitely slowly, and its parameter $t$ is not to be identified with real time. For example, a gas in a cylinder with a piston attached will undergo a quasistatic transition if the piston is withdrawn so slowly that the effect on the gas is reversible. Ifthe piston is withdrawn rapidly the action is irreversible, as non-equilibrium intermediate states arise in which the gas swirls and eddies, creating regions of non-uniform pressure and density throughout the container. The same can be said of the action of a ‘stirrer’ on a gas or liquid in an adiabatic container – you can never ‘unstir’ the milk or sugar added to a cup of tea. Irreversible transitions from one state of the system cannot be represented by parametrized curves in the manifold of equilibrium states $K$. Whether the transition be reversible or irreversible, we assume that there is always associated with it a well-defined quantity $\Delta W$ , known as the work done by the system. The work done on the system is defined to be the negative of this quantity, $-\Delta W$

We will also think of thermodynamic systems as being confined to certain ‘enclosures’, to be thought of as closed regions of three-dimensional space. Most importantly, a system $K$ is said to be in an adiabatic enclosure if equilibrium states can only be disturbed by doing work on the system through mechanical means (reversible or irreversible), such as the movement of a piston or the rotation of a stirrer. In all cases, transitions between states of a system in an adiabatic enclosure are called adiabatic processes.

The boundary of an adiabatic enclosure can be considered as being an insulating wall through which no ‘heat transfer’ is allowed; a precise meaning to the concept of heat wil be given directly. A diathermic wall within an adiabatic enclosure is one that permits heat to be transferred across it without any work being done. Two systems $K_A$ and $K_B$ are said to be in thermal contact if both are enclosed in a common adiabatic enclosure, but are separated by a diathermic wall. The states ${\mathbf{X}}_A$ and ${\mathbf{X}}_B$ of the two systems are then said to be in thermal equilibrium with each other.

Zeroth law of thermodynamics: temperature For every thermodynamic system $K$ there exists a function $\tau :KR$ called empirical temperature such that two systems $K_A$ and $K_B$ are in equilibrium with each other if and only $if{\tau }_A({\mathbf{x}}_A)={\tau }_B({\mathbf{x}}_B)$

This law serves as little more than a definition of empirical temperature, but the fact that a single function of state achieves the definition of equilibrium is significant. Any set of states $\{\mathbf{x}\mid \tau (\mathbf{x})=const.\}$ is called an isotherm of a system $K$

Example 16.3

For an ideal gas we find $\tau =pV$ is an empirical temperature, and the isotherms are curves $pV=\mathrm{const}.$ . Any monotone function ${\tau }^{\prime }=\phi \circ \tau$ will also do as empirical temperature. A system of ideal gases in equilibrium with each other, $(p_1,V_1),(p_2,V_2),...,(p_n,V_n)$ have common empirical temperature $\tau =T_g$ , called the absolute gas temperature, given by

$$T_g=\frac{p_1{V}_1}{n_{1}R}=\frac{p_2{V}_2}{n_{2}R}=\cdots =\frac{p_n{V}_n}{n_{n}R}$$

where $n_i$ are the relative molar quantities of the gases involved and $R$ is the universal gas constant. While an arbitrary function $\phi$ may still be applied to the absolute gas temperature, the same function must be applied equally to all component gases. In this example it is possible to eliminate all pressures except one, and the total system can be described by a single thermal variable, $p_1$ say, and $n$ external deformation variables $V_1,V_2,\dots ,V_n$

This example illustrates a common assumption made about thermodynamic systems of $n$ degrees of freedom, that it is possible to pick coordinates $(x^1,x^2,\dots ,x^n)$ in a local neighbourhood ofany point in $K$ such that the first $n-1$ coordinates are external variables and $x^n$ is an internal variable. We call this the thermal variable assumption.

First law of thermodynamics: energy. For every thermodynamic system $K$ there is a function $U:K\mathbb{R}$ known as internal energy and a 1-form $\omega \in {\Lambda }_1(K)$ known as the workform such that the work done by the system in any reversibleprocess $\gamma :[t_1,t_2]K$ is given by

$$\Delta W={\int }_{\gamma }\omega$$

(see Example 15.9for the definition ofintegral of$\omega$ along the curve $\gamma$). In every reversible adiabatic process

$${\gamma }^{\ast }(\omega +\mathrm{d}U)=0.$$

From Example 15.9 the integral of $\omega +\mathrm{d}U$ along the curve $\gamma$ vanishes, since

$${\int }_{\gamma }\omega +\mathrm{d}U={\int }_{t_1}^{t_2}⟨{\gamma }^{\ast }(\omega +\mathrm{d}U),\frac{\mathrm{d}}{\mathrm{d}t}⟩\mathrm{d}t=0.$$

Furthermore, since

$${\int }_{\gamma }\mathrm{d}U={\int }_{t_1}^{t_2}\frac{\mathrm{d}U}{\mathrm{d}t}\mathrm{d}t=U(t_2)-U(t_1)=\Delta U$$

the conservation law of energy holds for any reversible adiabatic process,

$$\Delta W+\Delta U=0.$$

Thus the change of internal energy is equal to the work done on the system, $-\Delta W$ . The work done in any reversible adiabatic transition from one equilibrium state of a system $K$ to another is independent of the path. In particular, no work is done by the system in any cyclic adiabatic process, returning a system to its original state – commonly known as the impossibility ofa perpetual motion machine of thefirst kind.

The heat 1-form is defined to be $\theta =\omega +\mathrm{d}U$ , and we refer to $\begin{array}{r}\Delta Q={\int }_{\gamma }\theta \end{array}$ as the heat added to the system in any reversible process $\gamma$. The conservation of energy in the form

$$\Delta Q=\Delta W+\Delta U$$

is often referred to in the literature as the first law of thermodynamics. Adiabatic transitions are those with $\Delta Q=0$

Ifthe thermal variable assumption holds, then it is generally assumed that the work form is a linear expansion of the external variables alone,

$$\omega =\sum _{k=1}^{n-1}{P}_k(x^1,\dots ,x^n)\mathrm{d}{x}^k,$$

where the component function $P_k$ is known as the $k$th generalized force. Since $U$ is a therma variable, it is always possible to choose the $n$th coordinate as $x^n=U$ , in which case

$$\theta =\sum _{k=1}^{n-1}{P}_k\left(x^1,\dots ,x^n\right)\mathrm{d}{x}^k+\mathrm{d}U=\sum _{k=1}^{n-1}{P}_k\left(x^1,\dots ,x^n\right)\mathrm{d}{x}^k+\mathrm{d}{x}^n.$$

Second law of thermodynamics

Not every transition between equilibrium states is possible, even if conservation of energy holds. The second law of thermodynamics limits the possible transitions consistent with energy conservation, and has a number ofequivalent formulations. For example, the version due to Clausis asserts that no machine can perform work, or mechanical energy, while at the same time having no other effect than to lower the temperature ofa thermodynamic system. Such a machine is sometimes referred to as aperpetual motion machine of the second kind – if it were possible one could draw on the essentially infinite heat reservoir of the oceans to perform an unlimited amount of mechanical work.

An equivalent version is Kelvin’s principle: no cyclic quasi-static thermodynamic process permits the conversion ofheat entirely into mechanical energy. By this is meant that no quasistatic thermodynamic cycle $\gamma$ exists, the first halfofwhich consists ofa quasi-static process ${\gamma }_1$ purely of heat transfer in which no work is done, ${\gamma }_1^{\ast }\omega =0$ , while the second half ${\gamma }_2$ is adiabatic and consists purely of mechanical work, ${\gamma }_2^{\ast }\theta =0$ . Since $U$ is a function of state it follows, on separating the cycle into its two parts, that

$$0=\Delta U={\oint }_{\gamma }\mathrm{d}U={\Delta }_{1}U+{\Delta }_{2}U={\Delta }_{1}Q-{\Delta }_{2}W.$$

Thus in any such cycle an amount of heat would be converted entirely into its mechanica equivalent of work.

Consider a quasi-static process ${\gamma }_1$ taking an equilibrium state $x$ to another state ${\mathbf{x}}^{\prime }$ along a curve of constant volume, $x^k=$ const. for $k=1,\dots ,n-1$ . Such a curve can be though of as ‘cooling at constant volume’ and is achieved purely by heat transfer; no mechanica work is done,

$${\gamma }_1^{\ast }\omega ={\gamma }_1^{\ast }\sum _{k=1}^{n-1}{P}_{k}d{x}^k=\sum _{k=1}^{n-1}{P}_{k}d\left(x^k\circ {\gamma }_1\right)=0.$$

It then follows that no reversible adiabatic transition ${\gamma }_2$ such that ${\gamma }_2^{\ast }\theta =0$ exists between these two states. Since processes such as ${\gamma }_1$ may always be assumed to be locally possible, it follows that every state $x$ has equilibrium states in its neighbourhood that cannot be reached by quasi-static adiabatic paths. This leads to Carathéodory’s more general version of the second law.

Second law of thermodynamics: entropy. In a thermodynamic system $K$, every neigh bourhood $U$ ofan arbitrary equilibrium state $x$ contains a state ${\mathbf{x}}^{\prime }$ that is inaccessible by a quasi-static adiabatic pathfrom $x$.

Theorem 16.5 · Carathéodory’s theorem

(Carathéodory) The heat 1-form $\theta$ is integrable, $\theta \wedge \mathrm{d}\theta =0$ , if and only if every neighbourhood of any state $\mathbf{x}\in K$ contains a state ${\mathbf{x}}^{\prime }$ adiabatically inaccessible from $x$.

Proof outline

If $\theta$ is integrable, then by Theorem 16.4 it is possible to find local co ordinates $(U;y^i)$ of any state $x$ such that $\theta {|}_U=Q_n\mathrm{d}{y}^n$ . Adiabatics satisfy ${\gamma }^{\ast }\theta =0$ , or $y^n=\mathbf{const}t$. Hence, if $U^{\prime }$ is an open neighbourhood of $x$ such that $U^{\prime }\subseteq U$ , any state ${\mathbf{x}}^{\prime }\in U^{\prime }$ such that $y^{\prime n}=y^n({\mathbf{x}}^{\prime })\ne y^n(\mathbf{x})$ is adiabatically inaccessible from $x$.

Conversely, if $\theta \wedge \mathrm{d}\theta \ne 0$ , then the 1-form $\theta$ is not integrable on an open subset $U$ of every state $\mathbf{x}\in M$ . Hence the distribution $D^{n-1}$ such that $\theta \in (D^{n-1}{)}^{\perp }$ is not involutive on an open neighbourhood $U^{\prime }$ of $x$, so that $[D^{n-1},D^{n-1}]=\mathcal{T}(U^{\prime })$ . Let $X$ and $Y$ be vector fields in $D^{n-1}$ such that $[X,Y]$ is not in the distribution. It may then be shown that every state ${\mathbf{x}}^{\prime }$ is accessible by a curve of the form

$$t\mapsto {\psi }_{-\sqrt{t}}\circ {\varphi }_{-\sqrt{t}}\circ {\psi }_{\sqrt{t}}\circ {\varphi }_{\sqrt{t}}\mathbf{x}$$

where ${\psi }_t$ and ${\varphi }_t$ are local flows generated by the vector field $X$ and $Y$ (see Example 15.14).

For a reversible adiabatic process $\gamma$ at constant volume we have ${\gamma }^{\ast }\theta =0$ and

$${\gamma }^{\ast }\omega =\sum _{k=1}^{n-1}{P}_k\frac{\mathrm{d}{x}^k(\gamma (t))}{\mathrm{d}t}\mathrm{d}t=0.$$

Hence there is no change in internal energy for such processes, $\Delta U=0$ . On the other hand, for an irreversible adiabatic process at constant volume, such as stirring a gas in an adiabatic enclosure, there is always an increase in internal energy, $U^{\prime }>U$ . Hence all states with $U^{\prime }<U$ are adiabatically inaccessible by adiabatic processes at constant volume, be they reversible or not. As remarked above, it is impossible to ‘unstir’ a gas. In general, for any two states $x$ and ${\mathbf{x}}^{\prime }$ either (i) $x$ is adiabatically inaccessible to ${\mathbf{x}}^{\prime },(\mathrm{ii}){\mathbf{x}}^{\prime }$ is adiabatically inaccessible to $x$, or (iii) there exists a reversible quasi-static process from $x$ and ${\mathbf{x}}^{\prime }$ .

From Theorem 16.5 and Carathéodory’s statement of the second law, the heat form $\theta$ can be expressed as

$$\theta =f\mathrm{d}s$$

where $f$ and $s$ are real-valued functions on $K$. Any function $s(x^1,\dots ,x^{n-1},U)$ for which this holds is known as an empirical entropy. A reversible adiabatic process $\gamma :[a,b]\to K$ is clearly isentropic, $s=const.$ , since ${\gamma }^{\ast }\theta =0$ along the process, and the hypersurface $s$ const. through any state $x$ represents the local boundary between adiabatically accessible and inaccessible states from .

For most thermodynamic systems the function $s$ is globally defined by the identity $\theta =f\mathrm{d}s$ . Since a path in $K$ connecting adiabatically accessible states has $\mathrm{d}U/\mathrm{d}t\ge 0$ , we can assume that $s$ is a monotone increasing function of $U$ for fixed volume coordinates $x^1,\dots ,x^{n-1}$ . For any path $\gamma$ with $x^k=\mathrm{const}.$ . for $k=1,\dots ,n-1$ , such that ${\gamma }^{\ast }\omega =0;$ , it follows that

$$\frac{\mathrm{d}U}{\mathrm{d}t}=⟨\dot{\gamma },\theta ⟩=f\frac{\mathrm{d}s}{\mathrm{d}t}$$

and the function $f$ must be everywhere positive.

Absolute entropy and temperature

Consider two systems $A$ and $B$ in an adiabatic enclosure and in equilibrium through mutua contact with a diathermic wall. In place of variables $x^1,\dots ,x^{n-1},U_A$ for states of system $A$ let us use variables $x^1,\dots ,x^{n-2},s_A,{\tau }_A$ where ${\tau }_A$ is the empirical temperature, and similarly use variables $y^1,\dots ,y^{m-2},s_B,{\tau }_B={\tau }_A$ for states of system $B$. The combined system then has coordinates $x^1,\dots ,x^{n-2},y^1,\dots ,y^{m-2},s_A,s_B,\tau ={\tau }_A={\tau }_B$ . Since work done in any reversible process is an additive quantity, $\Delta \mathbf{W}=\Delta {\mathbf{W}}_A+\Delta {\mathbf{W}}_B$ , we may assume from the first law of thermodynamics that $U$ is an additive function, $U=U_A+U_B$ . Hence the work 1-form may be assumed to be additive, $\omega ={\omega }_A+{\omega }_B$ , and so is the heat 1-form

$$\theta =\omega +\mathrm{d}U={\omega }_A+{\omega }_B+\mathrm{d}{U}_A+\mathrm{d}{U}_B={\theta }_A+{\theta }_B,\text{\hspace{1em}(16.15)}$$

which can be written

$$f\mathrm{d}s=f_A\mathrm{d}{s}_A+f_B\mathrm{d}{s}_B,\text{\hspace{1em}(16.16)}$$

where $f_A=f_A(x^1,\dots ,x^{n-2},s_A,\tau )$ and $f_B=f_B(y^1,\dots ,y^{m-2},s_B,\tau )$ . Since $s$ is a function of all variables $s=s(x^1,\dots ,y^{m-2},s_A,s_B,\tau )$ , it follows that $s=s(s_A,s_B)$ and

$$\frac{f_A}{f}=\frac{\partial s}{\partial s_A},\frac{f_B}{f}=\frac{\partial s}{\partial s_B}.\text{\hspace{1em}(16.17)}$$

Hence $f=f(s_A,s_B,\tau ),f_A=f_A(s_A,\tau ),f_B=f_B(s_B,\tau )\mathrm{and}$

$$\frac{\partial \ln f_A}{\partial \tau }=\frac{\partial \ln f_B}{\partial \tau }=\frac{\partial \ln f}{\partial \tau }=g(\tau )$$

for some function $g$. Setting $\begin{array}{r}T(\tau )=\exp \left(\int g(\tau )\mathrm{d}\tau \right)\end{array}$

$$f_A=T(\tau )F_A(s_A),f_B=T(\tau )F_B(s_B),f=T(\tau )F(s_A,s_B),$$

and Eq. (16.16) results in

$$F\mathrm{d}s=F_A\mathrm{d}{s}_A+F_B\mathrm{d}{s}_B.\text{\hspace{1em}(16.18)}$$

By setting $\begin{array}{r}{S}_A=\int F_A(s_A)\mathrm{d}{s}_A\end{array}$ and $\begin{array}{r}{S}_B=\int F_B({\mathbf{s}}_B)\mathrm{d}{\mathbf{s}}_B\end{array}$ , we have

$$F\mathrm{d}s=\mathrm{d}{S}_A+\mathrm{d}{S}_B=\mathrm{d}S$$

where $S=S_A+S_B$ . Hence

$${\theta }_A=f_A\mathrm{d}{s}_A=T{F}_A\mathrm{d}{s}_A=T\mathrm{d}{S}_A,{\theta }_B=T\mathrm{d}{S}_B$$

and

$$\theta =TF\mathrm{d}s=T\mathrm{d}S,\text{\hspace{1em}(16.19)}$$

which is consistent with the earlier requirement of additivity of heat forms, Eq. (16.15). The particular choice of empirical temperature $T$ and entropy $S$ such that Eq. (16.19) holds, and which has the additivity property $S=S_A+S_B$ , is called absolute temperature and absolute entropy. In the literature one often finds the formula $dQ=T\mathrm{d}S$ in place ofEq. (16.19) but this notation is not good, for the right-hand side is not an exact differential as $\mathrm{d}\theta \ne 0$ in general.

When $\mathrm{d}\theta \ne 0$ the original variables $\tau$ and $s$ are independent and only simple scaling freedoms are available for absolute temperature and entropy. For example, if

$$\theta =T\mathrm{d}S=T^{\prime }\mathrm{d}{S}^{\prime }$$

then

$$\frac{T^{\prime }(\tau )}{T(\tau )}=\frac{\mathrm{d}{S}^{\prime }(s)}{\mathrm{d}S(s)}=a=\text{ const. },$$

where $a>0$ if the rule $\Delta S>0$ for adiabatically accessible states is to be preserved. Hence

$$T^{\prime }=aT,S^{\prime }=\frac{1}{a}S+b.$$

Only a positive scaling may be applied to absolute temperature and there is an absolute zero of temperature; absolute entropy permits an affine transformation, consisting of both a rescaling and change of origin.

Example 16.4

An ideal or perfect gas is determined by two variables, volume $V$ and absolute temperature $T$. The heat 1-form is given by

$$\theta =\mathrm{d}U+p\mathrm{d}V=T\mathrm{d}S.$$

Using

$$\mathrm{d}\left(\frac{\theta }{T}\right)={\mathrm{d}}^{2}S=0$$

we have

$$-\frac{1}{T^2}\mathrm{d}T\wedge \mathrm{d}U+\mathrm{d}\left(\frac{p}{T}\right)\wedge \mathrm{d}V=0$$

and setting $U=U(V,T),p=p(V,T)$ results in

$$[-\frac{1}{T^2}(\frac{\partial U}{\partial V}{)}_T-\frac{p}{T^2}+\frac{1}{T}\left(\frac{\partial p}{\partial T}{)}_V\right]\mathrm{d}T\wedge \mathrm{d}V=0.$$

Hence

$$T{\left(\frac{\partial p}{\partial T}\right)}_V={\left(\frac{\partial U}{\partial V}\right)}_T+p.\text{\hspace{1em}(16.20)}$$

For a gas in an adiabatic enclosure, classic experiments of Gay-Lussac and Joule have led to the conclusion that $U=U(T)$ . Substituting into Eq. (16.20) results in

$$\frac{\partial \ln p}{\partial T}=\frac{1}{T},$$

which integrates to give a function $f(V)$ such that

$$f(V)p=T.$$

Comparing with the discussion in Example 16.3, we have for a single mole of gas

$$Vp=R{T}_g,$$

and since $T=T(T_g)$ it follows that after a suitable scaling of temperature we may set $f(V)=V$ and $T=T_g$ . Thus for an ideal gas the absolute temperature is identical with absolute gas temperature.

From $\theta =T\mathrm{d}S=\mathrm{d}U+p\mathrm{d}V$ we have

$$\mathrm{d}S=\frac{1}{T}\mathrm{d}U+\frac{R}{V}\mathrm{d}V$$

and the formula for absolute entropy of an ideal gas is

$$S=\int \frac{1}{T}\frac{\mathrm{d}U}{\mathrm{d}T}\mathrm{d}T+R\ln V.$$

Problem

Problem 16.9

For a reversible process $\sigma :TK$ , using absolute temperature $T$ as the parameter, set

$${\sigma }^{\ast }\theta =c\mathrm{d}T$$

where $c$ is known as the specific heat for the process. For a perfect gas show that for a process at constant volume, $V=const.$ ., the specific heat is given by

$$c_V={\left(\frac{\partial U}{\partial T}\right)}_V.$$

For a process at constant pressure show that

$$c_p=c_V+R,$$

while for an adiabatic process, ${\sigma }^{\ast }\theta =0.$

$$p{V}^{\gamma }=\text{ const. }\text{ where }\gamma =\frac{c_p}{c_V}.$$