Question

Under \[3{\text{ atm}}\], \[12.5{\text{ litre}}\] of a certain gas weight \[15{\text{ g}}\], calculate the average speed of gaseous molecules
A.\[7 \times {10^4}{\text{cm se}}{{\text{c}}^{{\text{ - 1}}}}\]
B.\[8.028 \times {10^4}{\text{cm se}}{{\text{c}}^{{\text{ - 1}}}}\]
C.\[6 \times {10^5}{\text{cm se}}{{\text{c}}^{{\text{ - 1}}}}\]
D.\[8.028 \times {10^6}{\text{cm se}}{{\text{c}}^{{\text{ - 1}}}}\]

Answer 1

Hint: To answer this question, you should recall the concept of the average speed of gas molecules. Study the dependence of factors which affect this average speed. Substitute the values in the formula (given below) to calculate the required answer.
The formula used:
\[{{\text{V}}_{{\text{A}}{\text{V}}}}{\text{}} = \sqrt {\dfrac{{8RT}}{{\pi m}}} \]
where \[{{\text{V}}_{{\text{A}}{\text{V}}}}\] = Root mean square speed, $R$ = Universal gas constant, $T$ = Temperature and $m$ is the Molar Mass of gas
\[PV = nRT\]
 where $P$ is pressure, $V$ is volume, $R$ is the universal gas constant, $n$ is no. of moles and $T$ is temperature

Complete step by step answer:
According to the Kinetic Molecular Theory of Gases, gas particles are in continuous motion and exhibit ideally elastic collisions.
In the question, we are given that
Pressure = \[3{\text{ atm}}\]; Volume \[ = {\text{ }}12.5{\text{ litre}}\] ; Weight = \[15{\text{ g}}\].
We can use the ideal gas equation to find the values of unknown variables in the formula of average speed. For gases we have the ideal gas equation:
\[PV = \dfrac{w}{m}RT\]
Substituting the values with appropriate units:
\[ \Rightarrow 3 \times 12.5\; = \;\dfrac{{15}}{m}RT\; \times 0.0821 \times T\].
We will arrive at\[\dfrac{T}{m}\; = {\text{ }}30.45\]
Now using this term and substituting this in the value of average speed:
\[{{\text{V}}_{{\text{A}}{\text{V}}}} = \sqrt {\dfrac{{8 \times 8.314 \times {{10}^7} \times 30.4 \times 7}}{{\;\pi }}} \]
After solving:
\[{{\text{V}}_{{\text{A}}{\text{V}}}} = \;8.028 \times {10^4}{\text{cm se}}{{\text{c}}^{{\text{ - 1}}}}\]

Hence, the correct answer to this question is option B.

Note:
Unless mentioned, we always assume the gas to obey the ideal gas equation. Along with different speeds, you should know the concept of the Maxwell-Boltzmann equation. The Maxwell-Boltzmann equation helps define the distribution of speeds for gas at various temperatures. From this distribution graph function, the most probable speed, the average speed, and the root-mean-square speed can be derived. The most probable speed is the speed most likely to be possessed by any molecule in the system.