Theory, worked examples and 24 previous-year questions with full solutions — IIT-JAM, GATE, JEST and CSIR NET-JRF, 2020 onwards. Click Show solution under any question.
is called exact. The variables \(c_i\) and \(x_i\) are said to be conjugate to each other. A general differential \(dF=M\,dx+N\,dy\) is exact if and only if
which expresses the equality of mixed second derivatives, \(\dfrac{\partial^2F}{\partial y\,\partial x}=\dfrac{\partial^2F}{\partial x\,\partial y}\). The integral of an exact differential depends only on the end points; around a closed path it is zero.
Partial-derivative identities
Let three variables be related by \(f(x,y,z)=0\), so any two can be chosen as independent.
Since \(dx\) and \(dz\) are independent, the coefficient of \(dx\) must be 1 (the reciprocal rule) and the coefficient of \(dz\) must be 0. Using the reciprocal rule on \((\partial x/\partial z)_y\) gives the cyclic rule.
Change of the variable held fixed. \(\Big(\dfrac{\partial x}{\partial y}\Big)_z=\Big(\dfrac{\partial x}{\partial y}\Big)_w+\Big(\dfrac{\partial x}{\partial w}\Big)_y\Big(\dfrac{\partial w}{\partial y}\Big)_z\).
2.2 Fundamental concepts
System, surroundings and boundary. A system is any object or region of space set apart (mentally) for study; everything else is the surroundings. The two are separated by a real or imaginary boundary. Thermodynamics deals with finite, macroscopic systems. A system is isolated if it exchanges neither matter nor energy with the surroundings, closed if it exchanges energy but not matter, and open if it exchanges both.
Thermodynamic state. The set of values of the macroscopic properties (\(T\), \(P\), \(V\), \(\rho\), …) that must be specified to reproduce the system.
State function. A variable that depends only on the present state, not on how it was reached. Its differential is exact, and its change is fully determined by the initial and final states. Examples: \(U\), \(S\), \(P\), \(V\), \(T\), \(H\), \(F\), \(G\).
Intensive and extensive variables. An intensive property does not depend on the size of the system (it is scale invariant); examples are \(P\), \(T\), density, chemical potential, viscosity, resistivity and specific heat capacity. An extensive property is additive for independent non-interacting subsystems, i.e. proportional to the amount of material; examples are \(U\), \(S\), \(G\), \(V\), mass, momentum and charge.
These properties combine according to simple rules. If \(f\) and \(g\) are intensive, then \(fg\), \(f/g\), \(df/dg\) and \(f+g\) are intensive. If \(F\) and \(G\) are extensive, then \(F+G\) is extensive, while \(F/G\) and \(dF/dG\) are intensive. Finally, if \(F\) is extensive and \(f\) is intensive, then \(fF\), \(F/f\) and \(dF/df\) are extensive.
2.3 The ideal gas
The ideal-gas law is the equation of state of a hypothetical gas. It follows from kinetic theory and contains Boyle’s and Charles’ laws:
Zeroth law. If two systems 1 and 2 are each in thermal equilibrium with a third system 3, they are in thermal equilibrium with each other. This justifies the concept of temperature.
First law. Energy is conserved when heat is taken into account. If an amount of heat \(\delta Q\) is supplied to a system and the system does work \(\delta W\), the change in its internal energy is
\[\boxed{dU=\delta Q-\delta W} .\]
Heat and work depend on the path, so \(\delta Q\) and \(\delta W\) are inexact differentials. By contrast, \(U\) is a state function, so \(dU\) is exact and \(\oint dU=0\). For a quasi-static process the work done by the system is \(\delta W=P\,dV\), and \(W_{A\to B}=\int_A^BP\,dV\) is the area under the path on a \(P\)–\(V\) diagram.
Sign convention. Work done by the system is positive; work done on it is negative. Heat absorbed is positive; heat rejected is negative.
Heat capacity
The heat capacity of a body is the heat needed to raise its temperature by one unit, \(C=\delta Q/dT\). The specific heat is the heat capacity per unit mass, \(c=\dfrac{\delta Q}{m\,dT}\), and the molar heat capacity is per mole.
Ideal gas. With \(f\) quadratic degrees of freedom, equipartition gives \(U=\frac f2N_Ak_BT=\frac f2RT\) per mole. So
Mayer’s relation (\(C_P-C_V=R\) for one mole of ideal gas). At constant pressure, \(\delta Q=dU+P\,dV\). For an ideal gas \(U=U(T)\), so \(dU=C_VdT\); also \(PV=RT\) gives \(P\,dV=R\,dT\) at constant \(P\). Hence \(C_P=\dfrac{\delta Q}{dT}\Big|_P=C_V+R\).
Response functions. The coefficient of volume expansion (expansivity) is \(\alpha=\dfrac1V\Big(\dfrac{\partial V}{\partial T}\Big)_P\), the isothermal elasticity (bulk modulus) is \(E_T=-V\Big(\dfrac{\partial P}{\partial V}\Big)_T\), and the isothermal compressibility is its reciprocal, \(\beta_T=\kappa_T=\dfrac{1}{E_T}=-\dfrac1V\Big(\dfrac{\partial V}{\partial P}\Big)_T\). For an ideal gas, \(\alpha=1/T\) and \(\kappa_T=1/P\). More generally, as is proved in Ch. 4, \(C_P-C_V=\dfrac{TV\alpha^2}{\kappa_T}\), \(\dfrac{\kappa_T}{\kappa_S}=\gamma\), and \(\dfrac{\alpha}{\kappa_T}=\Big(\dfrac{\partial P}{\partial T}\Big)_V\).
Example 2.1 Find the isothermal compressibility \(\kappa\) of a van der Waals gas as a function of volume \(V\) at temperature \(T\). (By definition, \(\kappa=-\frac1V\left(\frac{\partial V}{\partial p}\right)_T\).)
Solution. For one mole, \(P=\dfrac{RT}{V-b}-\dfrac{a}{V^2}\). Differentiating at constant \(T\) gives \(\Big(\dfrac{\partial P}{\partial V}\Big)_T=-\dfrac{RT}{(V-b)^2}+\dfrac{2a}{V^3}=-\dfrac{RTV^3-2a(V-b)^2}{V^3(V-b)^2}\). Hence
As a check, for \(a=b=0\) this gives \(\kappa=V/RT=1/P\), the ideal-gas value.
2.5 Thermodynamic processes and the first law
Throughout, \(n\) moles of an ideal gas go from state A to state B.
(a) Isochoric process (\(V\) constant). Here \(dV=0\), so \(W=0\) and \(\delta Q=dU=nC_VdT\): all the heat goes into internal energy.
(b) Isobaric process (\(P\) constant). The work done is \(W=\displaystyle\int_A^BP\,dV=P(V_B-V_A)=nR(T_B-T_A)\), and the heat absorbed is \(\delta Q=dU+P\,dV=nC_V\Delta T+nR\Delta T=n(C_V+R)\Delta T=nC_P\Delta T\).
(c) Isothermal process (\(T\) constant). For an ideal gas \(\Delta U=0\). The work done is \(W=\displaystyle\int_{V_A}^{V_B}\frac{nRT}{V}dV=nRT\ln\frac{V_B}{V_A}\), and hence \(Q=W=nRT\ln\dfrac{V_B}{V_A}\).
(d) Adiabatic process (\(\delta Q=0\)).
Derivation of \(PV^\gamma=\) const. With \(\delta Q=0\) the first law gives \(nC_VdT+P\,dV=0\), while from \(PV=nRT\) we have \(n\,dT=\dfrac{P\,dV+V\,dP}{R}\). Substituting, \(\dfrac{C_V}{R}(P\,dV+V\,dP)+P\,dV=0\). Multiplying by \(R/C_V\) and using \(R/C_V=\gamma-1\) yields \(\gamma P\,dV+V\,dP=0\). Integrating gives \(PV^\gamma=\) const, or in the equivalent forms \(TV^{\gamma-1}=\) const and \(P^{1-\gamma}T^\gamma=\) const.
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