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 obtained 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:
\[mF(x,y,z)=x\frac{\partial F}{\partial x}+y\frac{\partial F}{\partial y}+z\frac{\partial F}{\partial z}\]
For an ideal gas it follows that: \(\gamma_p=1/T\), \(\kappa_T=1/p\) and \(\beta_V=-1/V\).
For an ideal gas holds: \(C_{mp}-C_{mV}=R\). 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 holds:
\[C_p-C_V=T\left(\frac{\partial p}{\partial T}\right)_{V}\cdot\left(\frac{\partial V}{\partial T}\right)_{p}=-T\left(\frac{\partial V}{\partial T}\right)_{p}^2\left(\frac{\partial p}{\partial V}\right)_{T}\geq0\]
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 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. In differential form this becomes: \(đ Q=dU+đW\), where \(đ\) means that the it is not a differential of a state function. For a quasi-static process: \(đW=pdV\). So for a reversible process: \(đQ=dU+pdV\).
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.
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:
\[dS\geq\frac{đQ}{T}\]
If the only processes occurring are reversible: \(dS=đQ_{\rm rev}/T\). So, the entropy difference after a reversible process is:
\[S_2-S_1=\int\limits_1^2 \frac{đQ_{\rm rev}}{T}\]
So, for a reversible cycle: \(\displaystyle\oint\frac{đQ_{\rm rev}}{T}=0\).
For an irreversible cycle: \(\displaystyle\oint\frac{đQ_{\rm irr}}{T}<0\).
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.
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: \(đW_{\rm rev}=-\gamma dA\), with \(\gamma\) the surface tension. From this follows:
\[\gamma=\left(\frac{\partial U}{\partial A}\right)_{S}=\left(\frac{\partial F}{\partial A}\right)_{T}\]
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: \(\displaystyle\xi=\frac{\mbox{Cold delivered}}{\mbox{Work added}}\)
For adiabatic processes: \(W=U_1-U_2\). For reversible adiabatic processes Poisson’s equation holds. With \(\gamma=C_p/C_V\) one gets that \(pV^\gamma=\)constant. Also: \(TV^{\gamma-1}=\)constant and \(T^\gamma p^{1-\gamma}=\)constant. Adiabats are steeper on a \(p\)-\(V\) diagram than isotherms because \(\gamma>1\).
Here: \(H_2-H_1=\int_1^2 C_pdT\). For a reversible isobaric process: \(H_2-H_1=Q_{\rm rev}\).
This is also called the Joule-Kelvin effect and is 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 keeps 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, applied in refrigerators.
The system undergoes a reversible cycle with 2 isothemics and 2 adiabats:
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 machine 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
Stirling’s cycle consists of 2 isotherms and 2 isochorics. The efficiency in the ideal case is the same as for a Carnot cycle.
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:
The minimal work needed to attain a certain state is: \(W_{\rm min}=-W_{\rm max}\).
Phase transitions are isothermic 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 transition heat of phase \(\beta\) 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 exist also phase transitions with \(r_{\beta\alpha}=0\). For those there will occur only be a discontinuity in the second derivatives of \(G_m\). These second-order transitions appear at organization phenomena.
A phase-change of the 3rd order, so with e.g. \([\partial^3 G_m/\partial T^3]_p\) non continuous arises e.g. when ferromagnetic iron changes to the paramagnetic state.
When the number of particles within a system changes this number becomes a third quantity of state. 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 Gibb's phase rule. .
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
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: \(\mu_i^\alpha=\mu_i^\beta=\mu_i^\gamma\).
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 for a discrete system the Maxwell-Boltzmann distribution. 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: \(\displaystyle S=kN+kN\ln\left(\frac{V(2\pi mkT)^{3/2}}{Nh^3}\right)\)
Thermodynamics can be applied to other systems than gases and liquids. To do this the term \(đW=pdV\) has to be replaced with the correct work term, like \(đW_{\rm rev}=-Fdl\) for the stretching of a wire, \(đW_{\rm rev}=-\gamma dA\) for the expansion of a soap bubble or \(đW_{\rm rev}=-BdM\) for a magnetic system.
A rotating, non-charged black hole has a temperature of \(T=\hbar c/8\pi km\). 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 \(\sim m^3\).