GATE 2026 CY – Question 62

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 · Numerical answer

The translational partition function of H2 in a 1 L vessel at 300 K is $y\times10^{27}$. The value of y (one decimal place) is _____. Given H mass 1.008 amu; 1 amu = $1.661\times10^{-27}$ kg; $h=6.626\times10^{-34}$ J s; $k=1.381\times10^{-23}$ J/K.

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Correct answer: 2.79 to 2.81

Explanation

The translational partition function of a molecule of mass $m$ in a volume $V$ is
$$q_{trans}=V\left(\frac{2\pi mk_BT}{h^2}\right)^{3/2}.$$

**Mass of an $H_2$ molecule:**
$$m=2\times1.008\times1.661\times10^{-27}=3.3486\times10^{-27}\text{ kg}.$$

**The thermal factor:**
$$\frac{2\pi mk_BT}{h^2}=\frac{2\pi\times3.3486\times10^{-27}\times1.381\times10^{-23}\times300}{(6.626\times10^{-34})^2}=\frac{8.717\times10^{-47}}{4.390\times10^{-67}}=1.9855\times10^{20}\ \text{m}^{-2}.$$

Raise it to the power 3/2:
$$(1.9855\times10^{20})^{3/2}=2.7976\times10^{30}\ \text{m}^{-3}.$$

**With $V=1$ L $=10^{-3}$ m³:**
$$q_{trans}=10^{-3}\times2.7976\times10^{30}=2.80\times10^{27}.$$

So $y=\mathbf{2.8}$.