Question

The pressure of a gas in a \[100mL\] container is \[200kPa\] and the average translational energy of each gas particle is \[6\times {{10}^{-21}}\] . Find the number of gas particles in the container. How many moles are there in the container?

Answer 1

Hint: To find the number of gas particles in the container, we must know the number of moles of gas present in it. We have been given the pressure and the volume for the gas and we know the value of the universal gas constant, so we can use the ideal gas equation to find the number of moles and then proceed to find the no. of gas particles.

Formula Used:
\[KE=\dfrac{3}{2}KT\]
\[PV=nRT\]
\[N=n\times {{N}_{A}}\]

Complete step by step answer:
To use the ideal gas equation, we must be aware of the temperature of the gas system. To this end, we have been given the average translational kinetic energy of each gas particle.
We know that average kinetic energy for gas particles is given as \[KE=\dfrac{3}{2}KT\] where \[K\] is the Boltzmann’s constant, and \[T\] is the temperature of the gas.
Hence we can say that \[T=\dfrac{2\times KE}{3\times K}\]
From the ideal gas equation, we can say that \[PV=nRT\]
Substituting the expression of temperature in the above equation, we get \[PV=nR\times \dfrac{2\times KE}{3\times K}\]
By transposing the terms to the left-hand side, we get that \[n=\dfrac{3}{2}\times \dfrac{P\times V\times K}{R\times KE}\]
Substituting the values of Boltzmann constant, the pressure, the volume, the average kinetic energy and the gas constant in the above obtained equation, we get
\[\begin{align}
  & n=\dfrac{3\times (200\times {{10}^{3}})\times (100\times {{10}^{-6}})\times (1.38\times {{10}^{-23}})}{2\times 8.314\times 6\times {{10}^{-21}}}\left( \because 1mL={{10}^{-6}}{{m}^{3}} \right) \\
 & \Rightarrow n=\dfrac{6\times 1.38\times {{10}^{-22}}}{2\times 8.314\times 6\times {{10}^{-21}}} \\
 & \Rightarrow n=8.3\times {{10}^{-3}}mol \\
\end{align}\]
We have thus obtained the number of moles of the gas. Now using Avogadro’s number we can easily find the number of particles of the gas
Number of particles of the gas \[(N)=n\times {{N}_{A}}\] where \[{{N}_{A}}\] is Avogadro's number.
\[\begin{align}
  & \Rightarrow N=8.3\times {{10}^{-3}}\times 6.023\times {{10}^{23}} \\
 & \Rightarrow N=5\times {{10}^{21}} \\
\end{align}\]

Note: We can use the relation between the total translational kinetic energy of the gas and the pressure and volume of the gas to find the number of particles of the gas. We would have to use the energy of one particle given to us and multiply it by the total number of particles to find the total energy.
\[E=\dfrac{3}{2}nRT=\dfrac{3}{2}PV\] and \[E=Ne\] are the two alternate relations that we can use to obtain our answer.