Question

A flask contains \[{{10}^{-3}}\,{{m}^{3}}\] gas. At a temperature the number of molecules of oxygen is \[3.0\times {{10}^{22}}\]. The mass of an oxygen molecule is \[5.3\times {{10}^{-26}}kg\] and at that temperature the rms velocity of molecules is \[400\dfrac{m}{s}\]. The pressure in \[\dfrac{N}{{{m}^{2}}}\] of the gas in the flask is

\[A.\,8.48\times {{10}^{4}}\]
\[B.\,2.87\times {{10}^{4}}\]
\[C.\,25.44\times {{10}^{4}}\]
\[D.\,12.72\times {{10}^{4}}\]

Answer 1

Hint: This is a direct question. Using the kinetic molecular theory relation between the pressure of the gas, volume of the gas, the mass of each molecule of the gas, the number of molecules and the mean square speed of the gas molecules, we can solve this problem.

Formula used:
\[PV=\dfrac{1}{3}mN{{c}^{2}}\]

Complete step by step answer:
The kinetic molecular theory relation.
\[PV=\dfrac{1}{3}mN{{c}^{2}}\]
Where P is the pressure of the gas, V is the volume of the gas, m is the mass of each molecule of the gas, N is the number of molecules and \[{{c}^{2}}\] is the mean square speed of the gas molecules.
From the data, we have the data as follows.
The volume of the gas, \[V={{10}^{-3}}{{m}^{3}}\]
The mass of each molecule of the gas, \[m=5.3\times {{10}^{-26}}kg\]
The number of molecules, \[N=3.0\times {{10}^{22}}\]
The mean square speed of the gas molecules, \[c=400\dfrac{m}{s}\]
All the above parameters are given with the SI units, so, no need to convert the units of any of the above parameters.
Firstly, rearrange the terms to obtain the equation in terms of the pressure of the gas. So, we have,
\[P=\dfrac{1}{3}\dfrac{mN{{c}^{2}}}{V}\]
Now, substitute the given values in the formula of the kinetic molecular theory relation. So, we get,
\[\begin{align}
  & P=\dfrac{1}{3}\dfrac{5.3\times {{10}^{-26}}\times 3.0\times {{10}^{22}}\times {{(400)}^{2}}}{{{10}^{-3}}} \\
 & \Rightarrow P=\dfrac{1}{3}\times 254400 \\
\end{align}\]
Continue further calculation.
\[\begin{align}
  & P=84800 \\
 & \Rightarrow P=8.48\times {{10}^{4}}\dfrac{N}{{{m}^{2}}} \\
\end{align}\]

As the value of the pressure of the gas in the flask in terms of \[\dfrac{N}{{{m}^{2}}}\] is \[8.48\times {{10}^{4}}\dfrac{N}{{{m}^{2}}}\],

Thus, the option (A) is correct.

Note:
The units of the parameters should be taken care of. As in this case, the pressure of the gas is asked, even, the volume of the gas can be asked, by giving the values of the other parameters. There is no need to convert the units of any of the above parameters, as all the above parameters are given with the SI units.