5.9: Chapter 5 Problems
An underlined problem number or problem-part letter indicates that the numerical answer appears in Appendix I.
5.1
Show that the enthalpy of a fixed amount of an ideal gas depends only on the temperature.
5.2
From concepts in this chapter, show that the heat capacities \(C_V\) and \(C_p\) of a fixed amount of an ideal gas are functions only of \(T\).
5.3
During the reversible expansion of a fixed amount of an ideal gas, each increment of heat is given by the expression \(\dq=C_V \dif T + (nRT/V)\dif V\) (Eq. 4.3.4).
(a) A necessary and sufficient condition for this expression to be an exact differential is that the reciprocity relation must be satisfied for the independent variables \(T\) and \(V\) (see Appendix F). Apply this test to show that the expression is not an exact differential, and that heat therefore is not a state function.
(b) By the same method, show that the entropy increment during the reversible expansion, given by the expression \(\dif S=\dq/T\), is an exact differential, so that entropy is a state function.
5.4
This problem illustrates how an expression for one of the thermodynamic potentials as a function of its natural variables contains the information needed to obtain expressions for the other thermodynamic potentials and many other state functions.
From statistical mechanical theory, a simple model for a hypothetical “hard-sphere” liquid (spherical molecules of finite size without attractive intermolecular forces) gives the following expression for the Helmholtz energy with its natural variables \(T\), \(V\), and \(n\) as the independent variables: \[ A = -nRT\ln\left[cT^{3/2}\left(\frac{V}{n}-b\right)\right] - nRT + na \] Here \(a\), \(b\), and \(c\) are constants. Derive expressions for the following state functions of this hypothetical liquid as functions of \(T\), \(V\), and \(n\).
(a) The entropy, \(S\)
(b) The pressure, \(p\)
(c) The chemical potential, \(\mu\)
(d) The internal energy, \(U\)
(e) The enthalpy, \(H\)
(f) The Gibbs energy, \(G\)
(g) The heat capacity at constant volume, \(C_V\)
(h) The heat capacity at constant pressure, \(C_p\) (hint: use the expression for \(p\) to solve for \(V\) as a function of \(T\), \(p\), and \(n\); then use \(H=U+pV\))
5.6
Use the data in Table 5.1 to evaluate \(\pd{S}{A\subs{s}}{T,p}\) at \(25\units{\(\degC\)}\), which is the rate at which the entropy changes with the area of the air–water interface at this temperature.
5.7
When an ordinary rubber band is hung from a clamp and stretched with constant downward force \(F\) by a weight attached to the bottom end, gentle heating is observed to cause the rubber band to contract in length. To keep the length \(l\) of the rubber band constant during heating, \(F\) must be increased. The stretching work is given by \(\dw'=F\dif l\). From this information, find the sign of the partial derivative \(\pd{T}{l}{S,p}\); then predict whether stretching of the rubber band will cause a heating or a cooling effect.
(Hint: make a Legendre transform of \(U\) whose total differential has the independent variables needed for the partial derivative, and write a reciprocity relation.)
You can check your prediction experimentally by touching a rubber band to the side of your face before and after you rapidly stretch it.