Question

$0.014kg$ of nitrogen is enclosed in a vessel at a temperature of ${27}^{\circ }C$. The amount of heat energy to be supplied to the gas to double the rms speed of its molecules is approximately equal to

• A. 6350 J
• B. 7350 J
• C. 8350 J
• D. 9350 J

Answer 1

The correct option is D

9350 J

The root mean square speed is related to absolute temperature $T$ as

$c_{rms}=\sqrt{\frac{3kT}{m}}$

For a given gas, $m$ is fixed. Therefore, $c_{rms}\propto \sqrt{T}$, Hence in order to double the root

Means square speed, the absolute temperature must be increased to four times the initial

Value. Initial temperature $T_1=273+27=300K$. Therefore, final temperature

$T_2=4T_1=1200K$.

Since the volume of the vessels is fixed. $\Delta V=0$. Hence the heat energy supplied to the gas

Does no work on the gas; it only increases the internal energy of the molecules.

The increase in internal energy is

$\Delta U=n{C}_v\Delta T$

Since nitrogen is diatomic $C_v=\frac{5R}{2}$. The number of moles of nitrogen in the vessel is

$n=\frac{massing}{molecularmass}=\frac{0.014\times {10}^3}{28}=9353J$.

$\therefore \Delta U=0.5\times \frac{5R}{2}\times (1200-300)$

$=\frac{0.5\times 5\times 8.314\times 900}{2}=9353J$.

So the correct choice is (d)