Question

At the back of this book is a table of thermodynamic data for selected substances at room temperature. Browse through the $C_P$values in this table, and check that you can account for most of them (approximately) using the equipartition theorem. Which values seem anomalous?

Short answer

Short Answer:

For monoatomic gases: $C_P=20.775\mathrm{J}·{\mathrm{C}}^{-1}$

For diatomic gases: $C_P=29.085\mathrm{J}·{\mathrm{C}}^{-1}$

For polyatomic molecules: $C_P=33.24\mathrm{J}·{\mathrm{C}}^{-1}$

For a solid: $C_P=24.93\mathrm{J}·{\mathrm{C}}^{-1}$

Step-by-step solution

Given Information:

Table of thermodynamic data for selected substances at room temperature.

Step 1:

$C_P=8.31\left(1+\frac{3}{2}\right)$

For an ideal gas at constant pressure

$C_P=nR\left(1+\frac{f}{2}\right)$

In SI units, $R=8.31.\mathrm{J}·{\mathrm{mol}}^{-1}·{\mathrm{K}}^{-1}$. So for one mole for an ideal gas:

$C_p=8.31\left(1+\frac{f}{2}\right)$

Because we have three transitional degrees of freedom in monatomic gases, f=3, the heat capacity of monatomic gases is:

$C_P=8.31\left(1+\frac{3}{2}\right)$

$C_P=20.775\mathrm{J}·{\mathrm{C}}^{-1}$

This is in great agreement with the monatomic gases hydrogen, argon, helium, and neon, according to the table in Schroeder's book's appendix.

Step 2:

We have three transitional degrees of freedom and two rotational degrees of freedom in diatomic gases (f=5). As a result, the heat capacity of diatomic gases is:

$C_P=8.31\left(1+\frac{6}{2}\right)$

$C_P=33.24\mathrm{J}·{\mathrm{C}}^{-1}$

which is quite close to the values for ${\mathrm{O}}_2,{\mathrm{N}}_2$and ${\mathrm{H}}_2$and $CO$but the heat capacity of ${\mathrm{Cl}}_2$is higher than this value.

Step 3:

With three transitional degrees of freedom, two rotational degrees of freedom, and one vibrational degree of freedom in a polyatomic molecule, f=0, the heat capacity of diatomic gases is:

$C_P=8.31\left(1+\frac{6}{2}\right)$

$C_P=33.24\mathrm{J}·{\mathrm{C}}^{-1}$

From the table, the heat capacity for carbon dioxide is $C_P=33.24\mathrm{J}·{\mathrm{C}}^{-1}$which is higher than $C_P=33.24\mathrm{J}·{\mathrm{C}}^{-1}$by 3.87

Step 4:

For most solids and liquids, the heat capacity at constant pressure is given by:

$C_P\approx 8.31\frac{f}{2}$

much then f=0 the heat capacity is therefore:

$C_P=8.31\frac{6}{2}$

$C_P=24.93\mathrm{J}·{\mathrm{C}}^{-1}$