Question

How many degrees of freedom are associated with 2 grams of He at NTP?
A) 3
B) $3 \cdot 01 \times {10^{23}}$
C) $9 \cdot 03 \times {10^{23}}$
D) 6

Answer 1

Firstly we calculate the number of moles in 2 gram of helium .
After that we calculate number of molecules of helium in given number of moles
Now we already know that there are 3 degrees of freedom corresponding to 1 molecule of a monatomic gas.
Finally to calculate the total number of degrees of freedom in a monatomic gas we multiply the number of molecules and degree of freedom of 1 monatomic gas.

Complete step by step process:
According to the question we have 2 gm of helium.
Number of moles =given mass of substance divided by molar mass.
$\therefore$we already know the molar mass of He is 4 amu
So, moles of He=$\dfrac{2}{4} = \dfrac{1}{2}$
Now to calculate moles into molecules we multiply moles with the Avogadro's number
Mathematically, $N = m \times {A_0}$ where N=number of molecules
M=number of moles
${A_o}$=Avogadro's number
So ,$N = 6 \cdot 02 \times {10^{^{23}}} \times \dfrac{1}{2}$
$N = 3 \cdot 01 \times {10^{23}}$
Hence total number of molecules in $\dfrac{1}{2}$moles of He is $3 \cdot 01 \times {10^{23}}$
Now total degree of freedom is equal to molecules multiply by degree of freedom of 1 monatomic gas
$\therefore$total degree of freedom =$3 \times 3 \cdot 01 \times {10^{23}}$
Total degree of freedom =$9 \cdot 03 \times {10^{23}}$.

Hence, option (C) is the best option.

Note: Degree of freedom, often abbreviated as df, is a concept that may be thought of as that part of the sample size n not otherwise allocated. Df is related to the sample number, usually to the number of observations for continuous data methods and to the number of categories for categorical data methods.