Energy and Entropy: Winter-2026
HW 5: Due Day 23 W8 D3

1. Analyze TS rectangle Consider a monatomic ideal gas that undergoes a 4-step cyclic process. On a \(T\)-\(S\) diagram, the four steps of the process trace out a rectangle. The cycle proceeds in the clockwise direction around the rectangular path. The four sides of the rectangle correspond to
   (A) Constant temperature, \(T_h\)
   (B) Constant entropy, \(S_h\)
   (C) Constant temperature, \(T_l\)
   (D) Constant entropy, \(S_l\)
   
   1. Make a \(T\)-\(S\) diagram that represents this cyclic process. Label the axes, the four steps (A through D), the direction of each process, and the key values of \(S\) and \(T\). The horizontal axis corresponds to which variable? Why?
   2. Does a clockwise path in \(T\)-\(S\) space correspond to a heat engine or a heat pump?
   3. Create a table, like the one below, and fill in all the values in terms of \(T_l\), \(T_h\), \(S_l\) and \(S_h\):

      Process	\(\Delta U\)	\(Q\)	\(W\)
      A
      B
      C
      D

   4. If this cycle corresponds to a heat engine, find the efficiency in terms of \(T_h\) and \(T_l\). Alternatively, if this cycle corresponds to a heat pump, find the coefficient of performance in terms of \(T_h\) and \(T_l\).
2. Using Gibbs Free Energy You are given the following Gibbs free energy: \begin{equation*} G=-k T N \ln \left(\frac{a T^{5 / 2}}{p}\right), \end{equation*} where \(a\) is a constant (whose dimensions make the argument of the logarithm dimensionless).
   1. Compute the entropy.
   2. Work out the heat capacity at constant pressure \(C_p\).
   3. Find the connection among \(V\), \(p\), \(N\), and \(T\), which is called the equation of state (Hint: find the volume as a partial derivative of the Gibbs free energy). Simplify the final expression as much as possible.
   4. Find the internal energy \(U\) from the expression for \(G\) that you were given in the main prompt. Simplify the final expression as much as possible.
3. Helmholtz Free Energy of a Van Der Waals Gas The Helmholtz free energy of a van der Waals (vdW) gas can be written as: \begin{equation*} F=-N k T\left\{1+\ln \left[\frac{(V-N b) T^{\frac{3}{2}}}{N}\right]\right\}-\frac{a N^{2}}{V} \end{equation*} Where \(a\) and \(b\) are constants.
   1. Derive the equation of state (relationship between \(p\), \(T\), and \(V\)) for this Helmholtz free energy.
      Hint: The starting equations for this problem include the thermodynamic identity, the definition of Helmholtz free energy, \(F=U-TS\), and math identities such as the overlord equation.
      Bonus point: Rearrange the vdW equation of state to highlight any similiarites with the ideal gas equation of state (\(pV=NkT\)). To highlight similarities, group together terms that have dimensions of pressure, group together terms that have dimension of volume, etc.
   2. Using your expression from part (a), sketch or plot \(p(V)\) at various fixed temperatures. The volume axis should include \(Nb\) up to \(6Nb\). Your plot can be dimensionless (i.e. \(V/Nb\) on the x axis). Select values of \(NkT\) and \(aN^2\) that give curves with different shapes. Can you create a minima in pressure near \(V = 2Nb\)?
4. Ideal gas internal energy In this problem, you will prove that the internal energy of an ideal gas depends on temperature, but not on volume, based soley on the ideal gas equation: \begin{align} pV &= Nk_BT \end{align} and of course your knowledge of thermodynamics. It's a pretty tricky proof, so I'll step you through it.
   1. To begin with, use the Helmholtz free energy \(F=U-TS\) to show that \begin{align} \left(\frac{\partial {U}}{\partial {V}}\right)_{T} &= -p + T\left(\frac{\partial {S}}{\partial {V}}\right)_{T} \end{align} for any material.
   2. Now show that for any material \begin{align} \left(\frac{\partial {S}}{\partial {V}}\right)_{T} &= \left(\frac{\partial {p}}{\partial {T}}\right)_{V}. \end{align}
   3. Finally, show that for an ideal gas \begin{align} \left(\frac{\partial {U}}{\partial {V}}\right)_{T} &= 0. \end{align} Remember that the only statement we can assume about the ideal gas is \(pV=Nk_BT\). We have not been given an expression for \(U\).
5. Non-Ideal Gas

   The equation of state of a gas that departs from ideality can be approximated by \[ p=\frac{NkT}{V}\left(1+\frac{NB_{2}(T)}{V}\right), \] where \(B_{2}\) is called the second virial coefficient. \(B_{2}\) is a function of \(T\), so it is usually written as \(B_{2}(T)\). The function \(B_{2}(T)\) increases monotonically with temperature. Find \(\left(\frac{\partial {U}}{\partial {V}}\right)_{T}\) and determine its sign.

6. Plastic Rod When stretched to a length \(L\) the tension force \(\tau\) in a plastic rod at temperature \(T\) is given by its Equation of State \begin{equation*} \tau = a T^{2} (L - L_{o}) \end{equation*} where \(a\) is a positive constant and \(L_{o}\) is the rod's unstretched length. For an unstretched rod (i.e. \(L = L_{o}\)) the heat capacity at constant length is \(C_{L}=bT\) where \(b\) is a constant. Knowing the internal energy at \(T_{o}, L_{o}\) (i.e. \(U(T_{o},L_{o})\)) find the internal energy \(U(T_{f},L_{f})\) at some other temperature \(T_{f}\) and length \(L_{f}\).

   1. (1 point) Write an expression for the exact differential \(dU\) in terms of \(dT\) and \(dL\) (we've been calling this type of expression an “overlord equation”).

   2. Show that the partial derivative \((\partial U / \partial L)_{T} = -aT^{2}(L-L_{o})\).

   3. Integrate \(dU\) very carefully in the \(T-L\) plane, keeping in mind that \(C_{L} = bT\) holds only at \(L=L_{o}\) to find \(U(T_f,L_f)-U(T_0,L_0)\).