Q.1.23 - An Introduction to Thermal Physics

Question

Calculate the total thermal energy in a liter of helium at room temperature and atmospheric pressure. Then repeat the calculation for a liter of air. 

Step-by-Step Solution

Answer

The total thermal energy in a liter is for helium$\text{U=151.987 J}$ and for air$\text{U=253.312 J}$.

Step 1. Given data.

A liter of helium at ambient temperature at the pressure of one atmosphere$\text{1 atm=101325 Pa}$.

Step 2. Explanation.

Whereas a monatomic gas, like helium, has only three translational degrees of freedom, it is the simplest instance.

$\text{U=}\frac{f}{2}NkT=\frac{3}{2}NkT$

Thermal energy,

$$\begin{gathered} \text{U=}\frac{3}{2}\mathrm{NkT} \\ =\frac{3}{2}\mathrm{PV} \\ =\frac{3}{2}\times 101325\times 1\times {10}^{-3} \\ =151.987 \mathrm{J} \end{gathered}$$

Step 3. Total thermal energy.

A rotation around the bond's axis isn't counted because it doesn't change the molecule's position.

$\text{f=5}$

$$\begin{gathered} \text{U=}\frac{f}{2}NkT \\ =\frac{5}{2}NkT \\ U=\frac{5}{2}PV \\ =\frac{5}{2}\times 101325\times 1\times {10}^{-3} \\ =253.312.\mathrm{J} \end{gathered}$$