States of Matter - Result Question 79
####80. A gas bulb of $1 L$ capacity contains $2.0 \times 10^{21}$ molecules of nitrogen exerting a pressure of $7.57 \times 10^{3} Nm^{-2}$. 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)
Show Answer
Answer:
Correct Answer: 80. $(1: 3)$
Solution:
- Number of moles $=\frac{2 \times 10^{21}}{6 \times 10^{23}}=0.33 \times 10^{-2}$
$$ \begin{aligned} & & p=7.57 \times 10^{3} Nm^{-2} \ \text { Now, } & p V & =n R T \ \Rightarrow & & T=\frac{p V}{n R}=\frac{7.57 \times 10^{3} \times 10^{-3}}{0.33 \times 10^{-2} \times 8.314}=276 K \ \Rightarrow & u _{rms}= & \sqrt{\frac{3 R T}{M}}=\sqrt{\frac{3 \times 8.314 \times 276}{28 \times 10^{-3}}} m s^{-1}=496 ms^{-1} \end{aligned} $$
Also, $\frac{u _{\text {mps }}}{u _{\text {rms }}}=0.82$
$\Rightarrow \quad u _{mps}=0.82 \times u _{rms}=0.82 \times 496 ms^{-1}=407 ms^{-1}$
