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 $7.57 \times 10^{3} \hspace {1mm} Nm^{-2}$. 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. $(407 {~ms}^{-1})$
Solution:
Number of moles $=\dfrac{2 \times 10^{21}}{6 \times 10^{23}}=0.33 \times 10^{-2}$
$ p=7.57 \times 10^{3} Nm^{-2} $
$\text { Now, } p V =n R T $
$\Rightarrow T=\dfrac{p V}{n R}=\dfrac{7.57 \times 10^{3} \times 10^{-3}}{0.33 \times 10^{-2} \times 8.314}=276 K $
$\Rightarrow u _{rms}= \sqrt{\dfrac{3 R T}{M}}=\sqrt{\dfrac{3 \times 8.314 \times 276}{28 \times 10^{-3}}} m s^{-1}=496 ms^{-1}$
Also, $\dfrac{u _{\text {mps }}}{u _{\text {rms }}}=0.82$
$\Rightarrow \quad u _{mps}=0.82 \times u _{rms}=0.82 \times 496 ms^{-1}=407 ms^{-1}$
BETA
