Use the Euler-Maclaurin summation formula (Problem 18-14) to...

Use the Euler-Maclaurin summation formula (Problem 18-14) to show that $$ q_{\mathrm{rot}}(T)=\frac{\pi^{1 / 2}}{\sigma}\left(\frac{T}{\Theta_{\mathrm{rot}}}\right)^{3 / 2}+\frac{1}{6}+O\left(\frac{\Theta_{\mathrm{rot}}}{T}\right) $$ for a spherical top molecule. Show that the correction to replacing Equation $18.53$ by an integral is about $1 \%$ for $\mathrm{CH}_{4}$ and $0.001 \%$ for $\mathrm{CCl}_{4}$ at $300 \mathrm{~K}$.

Use the Euler-Maclaurin summation formula (Problem 18-14) to show that
$$
q_{\mathrm{rot}}(T)=\frac{\pi^{1 / 2}}{\sigma}\left(\frac{T}{\Theta_{\mathrm{rot}}}\right)^{3 / 2}+\frac{1}{6}+O\left(\frac{\Theta_{\mathrm{rot}}}{T}\right)
$$
for a spherical top molecule. Show that the correction to replacing Equation $18.53$ by an integral is about $1 \%$ for $\mathrm{CH}_{4}$ and $0.001 \%$ for $\mathrm{CCl}_{4}$ at $300 \mathrm{~K}$.

Key Concepts

Euler-Maclaurin Summation Formula

The Euler-Maclaurin summation formula provides a powerful method for approximating finite sums with integrals by adding correction terms involving higher derivatives of the function. This formula is particularly useful in bridging discrete sums and continuous integrals, allowing for asymptotic expansions and error estimates when simplifying the sum of a function over integer values.

Partition Function in Statistical Mechanics

The partition function is a fundamental quantity in statistical mechanics that encapsulates the statistical properties of a system in thermodynamic equilibrium. For rotational degrees of freedom, the partition function sums over the quantized rotational energy levels, and its evaluation (sometimes approximated by an integral) is crucial for determining thermodynamic quantities such as energy, entropy, and heat capacity.

Rotational Energy Levels of Spherical Top Molecules

Spherical top molecules have symmetric moments of inertia and exhibit degenerate rotational energy levels. Their rotational partition function is derived by summing over these energy states. Due to the symmetry, the rotational energy spectrum follows specific patterns that are key to calculating thermodynamic properties, particularly when comparing discrete sums to their continuous integral approximations.

Asymptotic Analysis and Integral Approximations

When dealing with large quantum numbers or high temperatures, the discrete rotational energy levels allow for a continuous approximation using integrals. Asymptotic analysis, such as invoking the Euler-Maclaurin formula, helps quantify the error introduced by replacing a sum with an integral and provides correction terms. This technique is essential in understanding the precision of approximations in practical calculations, including percentage corrections for specific molecules like CH? and CCl?.

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