Physics 731 – HW 1
Due Friday Sep 11, 2026
Problem 1: Temperature Scale: Prove the equivalence of ideal gas temperature scale Θ and
thermodynamic scale T by performing a Carnot cycle on an ideal gas. The ideal gas satisfies
P V = N kB Θ and E is a function of Θ. However, you may not assume E ∝ Θ.
Problem 2: Equations of State: the equation of state constrains the form of internal energy as in
the following examples.
• Starting from dE = T ds − P dV show that the equation of state P V = N kB T implies that E
can only depend on T .
• What is the most general equation of state consistent with an internal energy that depends
only on temperature.
• Show that for a van der Waals gas CV is a function of temperature alone.
Problem 3: Hard core gas: a gas obeys the equation of state P (V − N b) = N kB T and has a heat
capacity CV that is independent of temperature. (Assume N is kept fixed)
• Find the Maxwell relation involving ∂S/∂V |T,N .
• By calculating dE(T, V ) show that E is a function of T (and N ) only.
C
• Show that γ ≡ CVp = 1 + NCkVB .
• Show that an adiabatic change satisfies the equation P (V − N b)γ = constant.
Problem 4 (Optional - Bonus Point): Describe Nernst Theorem and discuss its implications.
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