Question

Calculate the heat capacity of liquid water per molecule, in terms of K . Suppose (incorrectly) that all the thermal energy of water is stored in quadratic degrees of freedom. How many degrees of freedom would each molecule have to have?

Short answer

Short Answer :

Heat capacity per molecule $C_m=9.072k$

Degrees of freedom f = 18

Step-by-step solution

Given Information :

Water's thermal energy is stored in quadratic degrees of freedom

Explanation 

The specific heat capacity of water is $c_{\text{water}}=1\mathrm{cal}·{\mathrm{g}}^{-1}·{\mathrm{C}}^{-1}=4.186{\mathrm{J}}^{-1}{\mathrm{g}}^{-1}·{\mathrm{C}}^{-1}$, and the molar weight of water is $18.015\mathrm{g}·$mol ${}^{-1}$so in $1\mathrm{gram}$of water.

$N=\frac{\text{Number of molecules in one mole}}{\text{Mass of one mole}}=\frac{6.022\times {10}^{23}}{18.015}$

$N=3.343\times {10}^{22}$molecules

The heat capacity of a single molecule is therefore:

$$\begin{gathered} C_m=\frac{C}{N}=\frac{4.186}{3.343\times {10}^{22}}=1.252\times {10}^{-22}\mathrm{J}·{\mathrm{C}}^{-1} \\ {C}_m=1.252\times {10}^{-22}\times \frac{k}{k}=1.252\times {10}^{-22}\times \frac{k}{1.38\times {10}^{-23}} \\ {C}_m=9.072k \end{gathered}$$

Explanation

The heat capacity is linearly proportional to the number of degrees of freedom, according to the equipartition theorem,

$C_m=f\frac{k}{2}$

Therefore now on comparing heat capacities, we have

$f\frac{k}{2}=9.072k$

$f=18$