Problem 82
Show that $$c_{v}=-T\left(\frac{\partial v}{\partial T}\right)_{s}\left(\frac{\partial P}{\partial T}\right)_{v} \text { and } c_{p}=T\left(\frac{\partial P}{\partial T}\right)_{s}\left(\frac{\partial v}{\partial T}\right)$$
Problem 83
Temperature and pressure may be defined as $$T=\left(\frac{\partial u}{\partial s}\right)_{v} \text { and } P=-\left(\frac{\partial u}{\partial v}\right).$$ Using these definitions, prove that for a simple compressible substance $$\left(\frac{\partial s}{\partial v}\right)_{u}=\frac{P}{T}.$$
Problem 84
For ideal gases, the development of the constant pressure specific heat yields $$\left(\frac{\partial h}{\partial P}\right)_{T}=0$$ Prove this by using the definitions of pressure and temperature, \(T=(\partial u / \partial s)_{v}\) and \(P=-(\partial u / \partial v)_{s}\).
Problem 86
For a homogeneous (single-phase) simple pure substance, the pressure and temperature are independent properties, and any property can be expressed as a function of these two properties. Taking \(v=v(P, T),\) show that the change in specific volume can be expressed in terms of the volume expansivity \(\beta\) and isothermal compressibility \(\alpha\) as \\[\frac{d v}{v}=\beta d T=\alpha d P\\] Also, assuming constant average values for \(\beta\) and \(\alpha,\) obtain a relation for the ratio of the specific volumes \(v_{2} / v_{1}\) as a homogeneous system undergoes a process from state 1 to state 2.
Problem 88
Consider an infinitesimal reversible adiabatic compression or expansion process. By taking \(s=s(P, v)\) and using the Maxwell relations, show that for this process \(P v^{k}=\) constant, where \(k\) is the isentropic expansion exponent defined as $$k=\frac{V}{P}\left(\frac{\partial P}{\partial V}\right)$$ Also, show that the isentropic expansion exponent \(k\) reduces to the specific heat ratio \(c_{p} / c_{v}\) for an ideal gas.
Problem 90
Steam is throttled from 2.5 MPa and \(400^{\circ} \mathrm{C}\) to 1.2 MPa. Estimate the temperature change of the steam during this process and the average Joule-Thomson coefficient.
Problem 91
The volume expansivity \(\beta\) values of copper at \(300 \mathrm{K}\) and \(500 \mathrm{K}\) are \(49.2 \times 10^{-6} \mathrm{K}^{-1}\) and \(54.2 \times 10^{-6} \mathrm{K}^{-1},\) respectively, and \(\beta\) varies almost linearly in this temperature range. Determine the percent change in the volume of a copper block as it is heated from \(300 \mathrm{K}\) to \(500 \mathrm{K}\) at atmospheric pressure.
Problem 92
The volume expansivity \(\beta\) values of copper at \(300 \mathrm{K}\) and \(500 \mathrm{K}\) are \(49.2 \times 10^{-6} \mathrm{K}^{-1}\) and \(54.2 \times 10^{-6} \mathrm{K}^{-1},\) respectively, and \(\beta\) varies almost linearly in this temperature range. Determine the percent change in the volume of a copper block as it is heated from \(300 \mathrm{K}\) to \(500 \mathrm{K}\) at atmospheric pressure.
Problem 94
Methane is to be adiabatically and reversibly compressed from 50 psia and \(100^{\circ} \mathrm{F}\) to 500 psia. Calculate the specific work required for this compression treating the methane as an ideal gas with variable specific heats and using the departure charts.
Problem 95
Refrigerant-134a undergoes an isothermal process at \(40^{\circ} \mathrm{C}\) from 2 to \(0.1 \mathrm{MPa}\) in a closed system. Determine the work done by the refrigerant-134a by using the tabular (EES) data and the generalized charts, in \(\mathrm{kJ} / \mathrm{kg}\).
