
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
135.6k+ views
Hint: 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.
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.
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