Problem 43
\(\text { Show that } c_{p}-c_{v}=T\left(\frac{\partial P}{\partial T}\right)_{v}\left(\frac{\partial v}{\partial T}\right)_{P}\).
Problem 46
Derive an expression for the isothermal compressibility of a substance whose equation of state is $$P=\frac{R T}{v-b}-\frac{a}{v(v+b) T^{1 / 2}}$$.where \(a\) and \(b\) are empirical constants.
Problem 47
Derive an expression for the volume expansivity of a substance whose equation of state is $$P=\frac{R T}{v-b}-\frac{a}{v^{2} T}$$ where \(a\) and \(b\) are empirical constants.
Problem 49
Demonstrate that $$k=\frac{c_{p}}{c_{v}}=-\frac{v \alpha}{(\partial v / \partial P)_{s}}$$
Problem 50
The Helmholtz function of a substance has the form $$a=-R T \ln \frac{v}{v_{0}}-c T_{0}\left(1-\frac{T}{T_{0}}+\frac{T}{T_{0}} \ln \frac{T}{T_{0}}\right)$$ where \(T_{0}\) and \(v_{0}\) are the temperature and specific volume at a reference state. Show how to obtain \(P, h, s, c_{v}\) and \(c_{p}\) from this expression.
Problem 51
Show that the enthalpy of an ideal gas is a function of temperature only and that for an incompressible substance it also depends on pressure.
Problem 52
What does the Joule-Thomson coefficient represent?
Problem 55
Does the Joule-Thomson coefficient of a substance shange with temperature at a fixed pressure?
Problem 56
Will the temperature of helium change if it is throttled adiabatically from \(300 \mathrm{K}\) and \(600 \mathrm{kPa}\) to \(150 \mathrm{kPa} ?\)
Problem 57
Estimate the Joule-Thomson coefficient of nitrogen at \((a) 120\) psia and \(350 \mathrm{R},\) and (b) 1200 psia and 700 R. Use nitrogen properties from EES or other source.
