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

Total thermal energy in a liter of helium : 151.98 J

Total thermal energy in a liter of air : 253.31.J

Step1: Given information

Temperature T = 298 K

Degree of freedom of Helium is f=3

Step2: Explanation

Total thermal energy is for a system contains N molecules each with f degree of freedom is given as 

$U_{\text{thermal }}=Nf\left(\frac{1}{2}kT\right) ..............................\left(1\right)$

Where 

N = number pf molecules

k = Boltzmann constants

f = degree of freedom 

T = Temperature 

A liter of helium at room temperature and pressure of atm = 101325 Pa

Using ideal gas law P V=n k T,  Find the thermal energy of

$$\begin{gathered} U_{\text{thermal }}=3NkT\frac{1}{2} \\ {U}_{\text{thermal }}=\frac{3}{2}PV..............................\left(2\right) \end{gathered}$$

Substitute the values given we get 

$$\begin{gathered} U_{\text{thermal }}=\frac{3}{2}PV \\ =\frac{3}{2}\times (101325 Pa)\times (1\times {10}^{-3}{m}^3) \\ {U}_{\text{thermal }}=151.98 \mathrm{J} \end{gathered}$$

For air , Number of degree of freedom is f=5
Find the thermal energy by substituting values in equation 2.

$$\begin{gathered} U_{\text{thermal }}=\frac{5}{2}PV \\ =\frac{5}{2}\times \left(101325Pa\right)\times (1\times {10}^{-3}{m}^3) \\ {U}_{\text{thermal }}=253.31 \mathrm{J} \end{gathered}$$