= Solution
For a <frozen ortho-para mixture>, the mole fractions do not change during the heat measurement, and the molar <heat capacity> is the composition-weighted average. A high-temperature composition retained on cooling has $x_o=3/4$, $x_p=1/4$, so
$$
\boxed{\frac{C_{V,m}^{\rm rot}}R=\tfrac14(1.425284)+\tfrac34(0.252645)
\simeq0.54581,\qquad C_{V,m}\simeq2.04581R.}
$$
For a mixture that \b[continues to interconvert and remain in equilibrium] near the specified <temperature>, use the combined rotational sum and differentiate that sum:
$$
q_{\rm eq}=q_{\rm even}+3q_{\rm odd}=4.075,\qquad
q_{\rm eq}'=0.02668\ {\rm K}^{-1},\qquad
q_{\rm eq}''=-5.66\times10^{-5}\ {\rm K}^{-2}.
$$
Consequently
$$
\boxed{\frac{C_{V,m,\rm eq}^{\rm rot}}R
=\frac{22500(-5.66\times10^{-5})}{4.075}
-22500\left(\frac{0.02668}{4.075}\right)^2
+\frac{300(0.02668)}{4.075}
\simeq0.68716,\qquad C_{V,m,\rm eq}\simeq2.18716R.}
$$
The equilibrium composition itself is $K=3(0.970)/1.165\simeq2.4979$, giving $x_o\simeq0.71411$. Its changing composition supplies a further positive <heat capacity>: warming converts some lower-energy para molecules into higher-energy ortho molecules. One can isolate this term as
$$
\frac{C_{V,m,\rm eq}^{\rm rot}}R
=x_p\frac{C_{V,m,p}^{\rm rot}}R+x_o\frac{C_{V,m,o}^{\rm rot}}R
+T^2x_px_o\left(\frac{q_{\rm odd}'}{q_{\rm odd}}-\frac{q_{\rm even}'}{q_{\rm even}}\right)^2.
$$
For clarity, if a mixture is first equilibrated at this <temperature> and then its composition is frozen during measurement, the conversion term is absent: its rotational value is $0.58789R$ and its total value is $2.08789R$. This is distinct from the <equilibrium heat capacity of interconverting states> even though the instantaneous composition is the same. An experiment must specify conversion activity and equilibration time; compare sufficiently slow <temperature> steps with a conversion catalyst against a composition-preserving sample, while keeping the same thermal background and checking that the sample has reached the stated <temperature>.
\Image[/past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2021/paper-2-hydrogen-heat-capacity.png]
{title=Original ideal-rotor heat capacities showing the difference between frozen normal hydrogen and continuously equilibrating hydrogen}
Back to article page
