8: Thermodynamics

Mathematical introduction

If there exists a relation \(f(x,y,z)=0\) between 3 variables, one can write: \(x=x(y,z)\), \(y=y(x,z)\) and \(z=z(x,y)\). The total differential \(dz\) of \(z\) is than given by:

\[dz=\left(\frac{\partial z}{\partial x}\right)_{y}dx+\left(\frac{\partial z}{\partial y}\right)_{x}dy\]

By writing this also for \(dx\) and \(dy\) it can be shown that

\[\left(\frac{\partial x}{\partial y}\right)_{z}\cdot\left(\frac{\partial y}{\partial z}\right)_{x}\cdot\left(\frac{\partial z}{\partial x}\right)_{y}=-1\]

Because \(dz\) is a total differential \(\oint dz=0\).

A homogeneous function of degree \(m\) obeys: \(\varepsilon^m F(x,y,z)=F(\varepsilon x,\varepsilon y,\varepsilon z)\). For such a function Euler’s theorem applies:

\[mF(x,y,z)=x\frac{\partial F}{\partial x}+y\frac{\partial F}{\partial y}+z\frac{\partial F}{\partial z}\]

Definitions

• The isochoric pressure coefficient: \(\displaystyle \beta_V=\frac{1}{p}\left(\frac{\partial p}{\partial T}\right)_{V}  \)
• The isothermal compressibility: \(\displaystyle \kappa_T=-\frac{1}{V}\left(\frac{\partial V}{\partial p}\right)_{T}\)
• The isobaric volume coefficient: \(\displaystyle \gamma_p=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{p}\)
• The adiabatic compressibility: \(\displaystyle \kappa_S=-\frac{1}{V}\left(\frac{\partial V}{\partial p}\right)_{S}\)

For an ideal gas it follows that: \(\gamma_p=1/T\), \(\kappa_T=1/p\) and \(\beta_V=-1/V\).

Thermal heat capacity

• The specific heat at constant \(X\) is: \(\displaystyle C_X=T\left(\frac{\partial S}{\partial T}\right)_{X}\)
• The specific heat at constant pressure: \(\displaystyle C_p=\left(\frac{\partial H}{\partial T}\right)_{p}\)
• The specific heat at constant volume: \(\displaystyle C_V=\left(\frac{\partial U}{\partial T}\right)_{V}\)

For an ideal gas : \(C_{mp}-C_{mV}=R\). Further, if the temperature is high enough to thermalize all internal rotational and vibrational degrees of freedom: \(C_V= \frac{1}{2} sR\). Hence \(C_p= \frac{1}{2} (s+2)R\). From their ratio it now follows that \(\gamma=(2+s)/s\). For a lower \(T\) one needs only to consider the thermalized degrees of freedom. For a Van der Waals gas: \(C_{mV}= \frac{1}{2} sR+ap/RT^2\).

In general:

\[C_p-C_V=T\left(\frac{\partial p}{\partial T}\right)_{V}\cdot\left(\frac{\partial V}{\partial T}\right)_{p}=-T\left(\frac{\partial V}{\partial T}\right)_{p}^2\left(\frac{\partial p}{\partial V}\right)_{T}\geq0\]

Because \((\partial p/\partial V)_T\) is always \(<0\), the following is always valid: \(C_p\geq C_V\). If the coefficient of expansion is 0, \(C_p=C_V\), and this is true also at \(T=0\)K.

The laws of thermodynamics

The zeroth law states that heat flows from higher to lower temperatures. The first law is the conservation of energy. For a closed system: \(Q=\Delta U+W\), where \(Q\) is the total added heat, \(W\) the work done and \(\Delta U\) the difference in the internal energy. Notice that the work is taken as the work done by the system on the surroundings. In differential form this becomes: \(d\hspace{-1ex}\rule[1.25ex]{2mm}{0.4pt} Q=dU+d\hspace{-1ex}\rule[1.25ex]{2mm}{0.4pt}W\), where \(d\hspace{-1ex}\rule[1.25ex]{2mm}{0.4pt}\) means that the it is not a differential of a state function. For a quasi-static process: \(d\hspace{-1ex}\rule[1.25ex]{2mm}{0.4pt}W=pdV\). So for a reversible process: \(d\hspace{-1ex}\rule[1.25ex]{2mm}{0.4pt}Q=dU+pdV\).

For an open (flowing) system the first law is: \(Q=\Delta H+W_{\rm i}+\Delta E_{\rm kin}+\Delta E_{\rm pot}\). One can extract an amount of work \(W_{\rm t}\) from the system or add \(W_{\rm t}=-W_{\rm i}\) to the system. In chemistry, one uses the opposite convention, that positive work is work done on the system.

The second law states: for a closed system there exists an additive quantity \(S\), called the entropy, the differential of which has the following property:

\[dS\geq\frac{d\hspace{-1ex}\rule[1.25ex]{2mm}{0.4pt}Q}{T}\]

If the only processes occurring are reversible: \(dS=d\hspace{-1ex}\rule[1.25ex]{2mm}{0.4pt}Q_{\rm rev}/T\). So, the entropy difference after a reversible process is:

\[S_2-S_1=\int\limits_1^2 \frac{d\hspace{-1ex}\rule[1.25ex]{2mm}{0.4pt}Q_{\rm rev}}{T}\]

For a reversible cycle: \(\displaystyle\oint\frac{d\hspace{-1ex}\rule[1.25ex]{2mm}{0.4pt}Q_{\rm rev}}{T}=0\).

For an irreversible cycle: \(\displaystyle\oint\frac{d\hspace{-1ex}\rule[1.25ex]{2mm}{0.4pt}Q_{\rm irr}}{T}<0\).

The third law of thermodynamics is (Nernst's law):

\[\lim_{T\rightarrow0}\left(\frac{\partial S}{\partial X}\right)_{T}=0\]

From this it can be concluded that the thermal heat capacity \(\rightarrow0\) if \(T\rightarrow0\), so absolute zero temperature cannot be reached by cooling through a finite number of steps.

State functions and Maxwell relations

The state functions and their differentials are:

Internal energy:	\(U\)	\(dU=TdS-pdV\)
Enthalpy:	\(H=U+pV\)	\(dH=TdS+Vdp\)
Free energy:	\(F=U-TS\)	\(dF=-SdT-pdV\)
Gibbs free energy:	\(G=H-TS\)	\(dG=-SdT+Vdp\)

From this one can derive Maxwell’s relations:

\[\left(\frac{\partial T}{\partial V}\right)_{S}=-\left(\frac{\partial p}{\partial S}\right)_{V}~,~~\left(\frac{\partial T}{\partial p}\right)_{S}=\left(\frac{\partial V}{\partial S}\right)_{p}~,~~ \left(\frac{\partial p}{\partial T}\right)_{V}=\left(\frac{\partial S}{\partial V}\right)_{T}~,~~\left(\frac{\partial V}{\partial T}\right)_{p}=-\left(\frac{\partial S}{\partial p}\right)_{T}\]

From the total differential and the definitions of \(C_V\) and \(C_p\) it can be derived that:

\[TdS=C_VdT+T\left(\frac{\partial p}{\partial T}\right)_{V}dV~~\mbox{and}~~TdS=C_pdT-T\left(\frac{\partial V}{\partial T}\right)_{p}dp\]

For an ideal gas:

\[S_m=C_V\ln\left(\frac{T}{T_0}\right)+R\ln\left(\frac{V}{V_0}\right)+S_{m0}~~\mbox{and}~~ S_m=C_p\ln\left(\frac{T}{T_0}\right)-R\ln\left(\frac{p}{p_0}\right)+S_{m0}'\]

Helmholtz’ equations are:

\[\left(\frac{\partial U}{\partial V}\right)_{T}=T\left(\frac{\partial p}{\partial T}\right)_{V}-p~~,~~\left(\frac{\partial H}{\partial p}\right)_{T}=V-T\left(\frac{\partial V}{\partial T}\right)_{p}\]

For a macroscopic surface: \(d\hspace{-1ex}\rule[1.25ex]{2mm}{0.4pt}W_{\rm rev}=-\gamma dA\), with \(\gamma\) the surface tension. From this follows: 

\[\gamma=\left(\frac{\partial U}{\partial A}\right)_{S}=\left(\frac{\partial F}{\partial A}\right)_{T}\]

Processes

The efficiency \(\eta\) of a process is given by: \(\displaystyle\eta=\frac{\mbox{Work done}}{\mbox{Heat added}}\)

The cold factor \(\xi\) of a cooling down process is given by: \(\displaystyle\xi=\frac{\mbox{Cold delivered}}{\mbox{Work added}}\)

Reversible adiabatic processes

For adiabatic processes: \(W=U_1-U_2\). For reversible adiabatic processes using Poisson’s equation with \(\gamma=C_p/C_V\) one gets that \(pV^\gamma=\)constant. Also: \(TV^{\gamma-1}=\)constant. Also \(T^\gamma p^{1-\gamma}=\)constant. Adiabats are steeper on a \(p\)-\(V\) diagram than isotherms because \(\gamma>1\).

Isobaric processes

Here: \(H_2-H_1=\int_1^2 C_pdT\). For a reversible isobaric process: \(H_2-H_1=Q_{\rm rev}\).

Throttle processes

This is also called the Joule-Kelvin effect and is the result of an adiabatic expansion of a gas through a porous material or a small opening. Here \(H\) is a conserved quantity, and \(dS>0\). In general this is accompanied with a change in temperature. The quantity which is important here is the throttle coefficient:

\[\alpha_H=\left(\frac{\partial T}{\partial p}\right)_{H}=\frac{1}{C_p}\left[T\left(\frac{\partial V}{\partial T}\right)_{p}-V\right]\]

The inversion temperature is the temperature where an adiabatically expanding gas maintains the same temperature. If \(T>T_{\rm i}\) the gas heats up, if \(T<T_{\rm i}\) the gas cools down. \(T_{\rm i}=2T_{\rm B}\), with for \(T_{\rm B}\): \([\partial(pV)/\partial p]_T=0\). The throttle process is, for example, used in refrigerators.

The Carnot Cycle

The system undergoes a reversible cycle consisting of two isotherms and two adiabats

1. Isothermal expansion at \(T_{1} \). The system absorbs an amount of heat \(Q_{1} \) from the reservoir.
2. Adiabatic expansion with a temperature drop to \(T_{2} \)
3. Isothermal compression at \(T_{2} \) removing an amount of heat \(Q_{2} \) from the system.
4. Adiabatic compression with the temperature increasing to \(T_{1} \)

The efficiency for a Carnot cycle is:

\[\eta=1-\frac{|Q_2|}{|Q_1|}=1-\frac{T_2}{T_1}:=\eta_{\rm C}\]

The Carnot efficiency \(\eta_{\rm C}\) is the maximal efficiency at which a heat engine can operate. If the process is applied in reverse order and the system performs a work \(-W\) the cold factor is given by:

\[\xi=\frac{|Q_2|}{W}=\frac{|Q_2|}{|Q_1|-|Q_2|}=\frac{T_2}{T_1-T_2}\]

The Stirling cycle

The Stirling cycle consists of 2 isotherms and 2 isochoric processes. The efficiency in the ideal case is the same as for a Carnot cycle.

Maximum work

Consider a system that changes from state 1 into state 2, with the temperature and pressure of the surroundings given by \(T_0\) and \(p_0\). The maximum work which can be obtained from this change is, when all processes are reversible:

1. Closed system: \(W_{\rm max}=(U_1-U_2)-T_0(S_1-S_2)+p_0(V_1-V_2)\).
2. Open system: \(W_{\rm max}=(H_1-H_2)-T_0(S_1-S_2)-\Delta E_{\rm kin}-\Delta E_{\rm pot}\).

The minimum work needed to attain a certain state is: \(W_{\rm min}=-W_{\rm max}\).

Phase transitions

Phase transitions are isothermal and isobaric, so \(dG=0\). When the phases are indicated by \(\alpha\), \(\beta\) and \(\gamma\): \(G_m^\alpha=G_m^\beta\) and

\[\Delta S_m=S_m^\alpha - S_m^\beta=\frac{r_{\beta\alpha}}{T_0}\]

where \(r_{\beta\alpha}\) is the heat of transition from \(\beta\) to phase \(\alpha\) and \(T_0\) is the transition temperature. The following holds: \(r_{\beta\alpha}=r_{\alpha\beta}\) and \(r_{\beta\alpha}=r_{\gamma\alpha}-r_{\gamma\beta}\). Further

\[S_m=\left(\frac{\partial G_m}{\partial T}\right)_{p}\]

so \(G\) has a kink in the transition point and the derivative is discontinuous. In a two phase system Clapeyron’s equation is valid:

\[\frac{dp}{dT}=\frac{S_m^\alpha-S_m^\beta}{V_m^\alpha-V_m^\beta}= \frac{r_{\beta\alpha}}{(V_m^\alpha-V_m^\beta)T}\]

For an ideal gas one finds for the vapor line at some distance from the critical point:

\[p=p_0{\rm e}^{-r_{\beta\alpha/RT}}\]

There also exist also phase transitions with \(r_{\beta\alpha}=0\). For those there will only be a discontinuity in the second derivatives of \(G_m\). These second-order transitions appear as organization phenomena.

A phase-change of the 3rd order, with e.g. \([\partial^3 G_m/\partial T^3]_p\) non continuous arises e.g. when ferromagnetic iron changes to the paramagnetic state.

Thermodynamic potential

When the number of particles within a system changes this number becomes a third state function. Because addition of matter usually takes place at constant \(p\) and \(T\), \(G\) is the relevant quantity. If a system has many components this becomes:

\[dG=-SdT+Vdp+\sum_i\mu_idn_i\] where \(\displaystyle\mu=\left(\frac{\partial G}{\partial n_i}\right)_{p,T,n_j}\)

is called the thermodynamic potential. This is a partial quantity. For \(V\):

\[V=\sum_{i=1}^c n_i\left(\frac{\partial V}{\partial n_i}\right)_{n_j,p,T}:=\sum_{i=1}^c n_i V_i\]

where \(V_i\) is the partial volume of component \(i\). The following holds:

\[\begin{aligned} V_m&=&\sum_i x_i V_i\\ 0&=&\sum_i x_i dV_i\end{aligned}\]

where \(x_i=n_i/n\) is the molar fraction of component \(i\). The molar volume of a mixture of two components can be a concave line in a \(V\)-\(x_2\) diagram: the mixing leads to a contraction of the volume

The thermodynamic potentials are not independent in a multiple-phase system. It can be derived that \(\sum\limits_i n_i d\mu_i=-SdT+Vdp\), this gives at constant \(p\) and \(T\): \(\sum\limits_i x_i d\mu_i=0\) (Gibbs-Duhmen).

Each component has as many \(\mu\)’s as there are phases. The number of free parameters in a system with \(c\) components and \(p\) different phases is given by \(f=c+2-p\) which is called the Gibbs phase rule. 

Ideal mixtures

For a mixture of \(n\) components (the index \(^0\) is the value for the pure component):

\[U_{\rm mixture}=\sum_i n_i U^0_i~~,~~H_{\rm mixture}=\sum_i n_i H^0_i~~,~~ S_{\rm mixture}=n\sum_i x_i S^0_i+\Delta S_{\rm mix}\]

where for ideal gases: \(\Delta S_{\rm mix}=-nR\sum\limits_i x_i\ln(x_i)\).

For the thermodynamic potentials: \(\mu_i=\mu_i^0+RT\ln(x_i)<\mu_i^0\). A mixture of two liquids is rarely ideal: this is usually only the case for chemically related components or isotopes. In spite of this Raoult’s law holds for the vapour pressure for many binary mixtures: \(p_i=x_ip^0_i=y_ip\). Here \(x_i\) is the fraction of the \(i\)th component in liquid phase and \(y_i\) the fraction of the \(i\)th component in gas phase.

A solution for one component in a second gives rise to an increase in the boiling point \(\Delta T_{\rm k}\) and a decrease of the freezing point \(\Delta T_{\rm s}\). For \(x_2\ll1\):

\[\Delta T_{\rm k}=\frac{RT_{\rm k}^2}{r_{\beta\alpha}}x_2~~,~~ \Delta T_{\rm s}=-\frac{RT_{\rm s}^2}{r_{\gamma\beta}}x_2\]

with \(r_{\beta\alpha}\) the heat of evaporation and \(r_{\gamma\beta}<0\) the melting heat. For the osmotic pressure \(\Pi\) of a solution: \(\Pi V_{m1}^0=x_2RT\).

These are called collegative properties

Conditions for equilibrium

When a system evolves towards equilibrium the only changes that are possible are those for which: \((dS)_{U,V}\geq0\) or \((dU)_{S,V}\leq0\) or \((dH)_{S,p}\leq0\) or \((dF)_{T,V}\leq0\) or \((dG)_{T,p}\leq0\). In equilibrium for each component: \(\mu_i^\alpha=\mu_i^\beta=\mu_i^\gamma\).

Statistical basis for thermodynamics

The number of possibilities \(P\) to distribute \(N\) particles on \(n\) possible energy levels, each with a \(g\)-fold degeneracy is called the thermodynamic probability and is given by:

\[P=N!\prod_i\frac{g_i^{n_i}}{n_i!}\]

The most probable distribution, that with the maximum value for \(P\), is the equilibrium state. When Stirling’s equation, \(\ln(n!)\approx n\ln(n)-n\) is used, one finds the Maxwell-Boltzmann distribution for a discrete system . The occupation numbers in equilibrium are then given by:

\[n_i=\frac{N}{Z}g_i\exp\left(-\frac{W_i}{kT}\right)\]

The state sum \(Z\) is a normalization constant, given by: \(Z=\sum\limits_ig_i\exp(-W_i/kT)\). For an ideal gas:

\[Z=\frac{V(2\pi mkT)^{3/2}}{h^3}\]

The entropy can then be defined as:  \(S=k\;ln\left ( P \right ) \). For a system in thermodynamic equilibrium this becomes:

\[S=\frac{U}{T}+kN\ln\left(\frac{Z}{N}\right)+kN\approx\frac{U}{T}+k\ln\left(\frac{Z^N}{N!}\right)\]

For an ideal gas, with \(U=\frac{3}{2}kT\) then: \(\displaystyle S=kN+kN\ln\left(\frac{V(2\pi mkT)^{3/2}}{Nh^3}\right)\)

Application to other systems

Thermodynamics can be applied to other systems than gases and liquids. To do this the term \(d\hspace{-1ex}\rule[1.25ex]{2mm}{0.4pt}W=pdV\) has to be replaced with the correct work term, for example \(d\hspace{-1ex}\rule[1.25ex]{2mm}{0.4pt}W_{\rm rev}=-Fdl\) for the stretching of a wire, \(d\hspace{-1ex}\rule[1.25ex]{2mm}{0.4pt}W_{\rm rev}=-\gamma dA\) for the expansion of a soap bubble or \(d\hspace{-1ex}\rule[1.25ex]{2mm}{0.4pt}W_{\rm rev}=-BdM\) for a magnetic system.

A rotating, non-charged black hole has a temperature \(T=\hbar c/8\pi km\). It has an entropy \(S=Akc^3/4\hbar\kappa\) with \(A\) the area of its event horizon. For a Schwarzschild black hole \(A\) is given by \(A=16\pi m^2\). Hawkings area theorem states that \(dA/dt\geq0\).

Hence, the lifetime of a black hole \(\sim m^3\).