States of Matter 1 Question 79
80. A gas bulb of $1 \mathrm{~L}$ capacity contains $2.0 \times 10^{21}$ molecules of nitrogen exerting a pressure of $7.57 \times 10^{3} \mathrm{Nm}^{-2}$. Calculate the root mean square (rms) speed and the temperature of the gas molecules. If the ratio of the most probable speed to root mean square speed is 0.82 , calculate the most probable speed for these molecules at this temperature.
(1993, 4M)
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Answer:
Correct Answer: 80. $(1: 3)$
Solution:
- Number of moles $=\frac{2 \times 10^{21}}{6 \times 10^{23}}=0.33 \times 10^{-2}$
$$ \begin{aligned} & & p=7.57 \times 10^{3} \mathrm{Nm}^{-2} \ \text { Now, } & p V & =n R T \ \Rightarrow & & T=\frac{p V}{n R}=\frac{7.57 \times 10^{3} \times 10^{-3}}{0.33 \times 10^{-2} \times 8.314}=276 \mathrm{~K} \ \Rightarrow & u_{\mathrm{rms}}= & \sqrt{\frac{3 R T}{M}}=\sqrt{\frac{3 \times 8.314 \times 276}{28 \times 10^{-3}}} \mathrm{~m} \mathrm{~s}^{-1}=496 \mathrm{~ms}^{-1} \end{aligned} $$
Also, $\frac{u_{\text {mps }}}{u_{\text {rms }}}=0.82$
$\Rightarrow \quad u_{\mathrm{mps}}=0.82 \times u_{\mathrm{rms}}=0.82 \times 496 \mathrm{~ms}^{-1}=407 \mathrm{~ms}^{-1}$
