Energy and Entropy: Winter-2026 HW 6 (updated 3:40pm Mon) : Due Day 20 Tues 10/22
Analyze TS rectangle
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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, \(S_l\)
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?
Does a clockwise path in \(T\)-\(S\) space correspond to a heat engine or a heat pump?
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
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 \(T_l\).
Helmholtz Free Energy of a Van Der Waals Gas
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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.
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.
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 \(V = 2Nb\)?