Question

The mass of 15 gram of nitrogen is enclosed in a vessel at 300 K. What heat must be supplied to it to double the ‘rms’ velocity of its molecules
A. 10J
B. 10KJ
C. ${{10}^{3}}J$
D. 100J

Answer 1

The root mean square (rms) velocity of the molecules of a given gas is directly proportional to the square root of the temperature. With this find the final temperature of the gas if the required heat is supplied. Then use the formula for the change in internal energy to find the heat that is to be supplied.

Formula used:
$Q=n{{C}_{V}}\left( {{T}_{2}}-{{T}_{1}} \right)$,
where Q is the heat supplied, n is the number of moles, ${{C}_{V}}$ is specific heat capacity at constant volume and ${{T}_{2}},{{T}_{1}}$ are final and initial temperatures.
$n=\dfrac{m}{M}$,
where m is the given mass and M is molar mass of the gas.
${{C}_{V}}=\dfrac{f}{2}R$,
where f is degrees of freedom and R is gas constant.

Complete step by step answer:
The root mean square (rms) velocity of the molecules of a given gas is directly proportional to the square root of the temperature.
i.e. ${{v}_{rms}}\propto \sqrt{T}$.
$\Rightarrow v_{rms}^{2}\propto T$
This means that if we double the rms velocity of the molecules, then the temperature of the gas will increase by four times. It is given that the gas is enclosed in a closed vessel. Therefore, its volume remains constant. This means that the heat is supplied to the gas at constant volume. Then the heat supplied to the gas is equal to,
$Q=n{{C}_{V}}\left( {{T}_{2}}-{{T}_{1}} \right)$ ….. (i)
Here, ${{T}_{1}}=300K$ and ${{T}_{2}}=4{{T}_{1}}=1200$.
Nitrogen is a diatomic linear gas. This means that each molecule of nitrogen gas has 5 degrees of freedom.
$\Rightarrow f=5$.
$\Rightarrow {{C}_{V}}=\dfrac{f}{2}R=\dfrac{5}{2}R$.
And the molar mass of nitrogen gas is 28 gram.
$\Rightarrow n=\dfrac{m}{M}=\dfrac{15}{28}mol$
The value of $R=8.3J{{K}^{-1}}$
Substitute the known values in (i).
$\Rightarrow Q=\left( \dfrac{15}{28} \right)\left( \dfrac{5}{2}\times 8.3 \right)\left( 1200-300 \right)$
$\therefore Q\approx 10000J=10KJ$.
This means that the heat that must be supplied to the gas to double its rms velocity is equal to 10KJ.

Hence, the correct option is B.

Note: We can also use the first law of thermodynamics. According to the law, $Q=W+\Delta U$, where Q is the heat supplied to the gas, W is the work done by the gas and $\Delta U$ is the change in its internal energy.
In this case, work done is zero because the volume of the gas is constant.
$\Rightarrow Q=\Delta U$
And $\Delta U=\dfrac{f}{2}nR\Delta T$.