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GATE 2025 CY – Question 47

Physical Chemistry · Statistical Thermodynamics: Micro-canonical, Canonical and Grand canonical ensembles, Boltzmann distribution, Partition functions and thermodynamic properties. Statistical mechanics of non-interacting systems, ideal monoatomic, diatomic gases, translational, rotational, vibrational and electronic partition functions. · 2 marks · Multiple select

An approximate partition function $Q(N,V,T)$ of a gas is given below. $$Q(N,V,T)=\frac1{N!}\left(\frac{2\pi mk_BT}{h^2}\right)^{3N/2}(V-Nb)^N.$$ The equation of state(s) for this gas is/are [Note: $b$ is a parameter independent of volume.]

  1. $P(V-Nb)=Nk_BT$
  2. $PV^{(N-b)}=k_BT$
  3. $PV=Nk_BT$
  4. $P(V-Nb)=Nk_B$

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Correct answer: (A) $P(V-Nb)=Nk_BT$

Explanation

In the canonical ensemble the pressure follows from the partition function:
$$P=k_BT\left(\frac{\partial\ln Q}{\partial V}\right)_{N,T}.$$

**Take the logarithm** of $Q=\dfrac1{N!}\left(\dfrac{2\pi mk_BT}{h^2}\right)^{3N/2}(V-Nb)^N$. Only the last factor depends on $V$:
$$\ln Q=\text{(terms independent of }V)+N\ln(V-Nb).$$

**Differentiate:**
$$\left(\frac{\partial\ln Q}{\partial V}\right)_{N,T}=\frac{N}{V-Nb}.$$

**Pressure:**
$$P=\frac{Nk_BT}{V-Nb}\;\Rightarrow\;P(V-Nb)=Nk_BT\quad(\text{option A}).$$

(This is the equation of state of a gas of hard spheres with excluded volume $b$ per particle.)