Why is the entropy of container A the greatest? Question


  9.35 Container A is filled with 1.0 mol of the atoms of an ideal monatomic gas. Container B has 1.0 mol of atoms bound together as diatomic molecules that are not vibrationally active. Container C has 1.0 mol of atoms bound together as diatomic molecules that are vibrationally active. The containers
  all start at $T_\mathrm{i}$ and the temperature is increased to $T_\mathrm{f}.$ Rank the containers in order of increasing change in entropy. Explain
  your reasoning.


Answer


   B &lt; C &lt; A


My Approach

I thought that you were supposed to assume that container C had the most number of atoms and therefore a greater entropy automatically.
An accurate summary of the relative molar heat capacities of the different species is given in another answer and recapitulated in the following table together with the total heat capacity for each gas:

$$\begin{array}{|case|Cv|} \hline \textrm{molecule} &amp; C_{vm}/R &amp; NC_{vm}/R \\ \hline  \mathrm{A} &amp; 3/2 &amp; 1.5 \\ \mathrm{B} &amp; 5/2 &amp; 1.25  \\\mathrm{C} &amp; 7/2 &amp; 1.75 \\ \hline \end{array}$$

where $N$ is the total amount of gas, which based on the problem described in the OP is either $\pu{1 mol}$ of atoms ($\mathrm{A}$) or $\pu{\frac{1}{2} mol}$ of homonuclear diatomic molecules ($\mathrm{B}$ and $\mathrm{C}$).

Since the entropy change for the isochoric temperature change is $\Delta S = C_v \log\left( \frac{T_2}{T_1} \right)$, it follows that $\Delta S_C &gt; \Delta S_A &gt; \Delta S_B$.