Q.1.43 - An Introduction to Thermal Physics

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?

Step-by-Step Solution

Answer

Short Answer :

Heat capacity per molecule $C_m=9.072k$

Degrees of freedom f = 18

Given Information :

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

Step 2 : 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:

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

 

Step 3 : 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

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

$f=18$