MHT CET · Physics · Kinetic Theory of Gases
If the absolute temperature of a gas is increased 5 times, the r.m.s. velocity of the gas molecules will be
- A 5 times
- B 10 times
- C 25 times
- D \(\sqrt{5}\) times
Answer & Solution
Correct Answer
(D) \(\sqrt{5}\) times
Step-by-step Solution
Detailed explanation
The mean kinetic energy \(\frac{1}{2} \mathrm{~m}<\mathrm{v}>^2\) of the gas molecule is proportional to the temperature \(\mathrm{T}\) :
\(<\mathrm{v}>2 \propto \mathrm{T}\)
\(\frac{\left\langle\mathrm{v}_2\right\rangle}{\left\langle\mathrm{v}_1\right\rangle}=\sqrt{\frac{\mathrm{T}_2}{\mathrm{~T}_1}}\)
\(\therefore\left\langle\mathrm{v}_2\right\rangle=\sqrt{5}\left\langle\mathrm{v}_1\right\rangle\)
\(<\mathrm{v}>2 \propto \mathrm{T}\)
\(\frac{\left\langle\mathrm{v}_2\right\rangle}{\left\langle\mathrm{v}_1\right\rangle}=\sqrt{\frac{\mathrm{T}_2}{\mathrm{~T}_1}}\)
\(\therefore\left\langle\mathrm{v}_2\right\rangle=\sqrt{5}\left\langle\mathrm{v}_1\right\rangle\)
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