Question

A gas is kept at ${P_0}$ pressures in one vessel if masses of all molecules are made half and speeds of each molecule is doubled, and then find the final pressure?

Answer 1

Hint:The initial pressure of the vessel is given. We will first assign variables to the initial and final masses and the velocities of the molecules. After obtaining the relationship between the initial and final masses and velocities of the molecules, we will apply these values in the formula for pressure of the molecules in the vessel and obtain the final pressure.

Complete step by step answer:
 Pressure is defined as the force acting upon a certain unit area. In our case, pressure in the vessel is the force acting upon the gas inside the vessel per area of the vessel. Here, it is given that the initial pressure of the vessel is given as ${P_0}$. Let the initial speeds of the molecules be ${v_1}$ and the initial masses of the molecules are ${M_1}$. Let the final speeds and masses of the molecules be ${v_2}$ and ${M_2}$. Thus, 2${M_2}$ = ${M_1}$ and ${v_2}$ = 2${v_1}$.

Now, the pressure of the vessel is given by the following formula as we all know;
\[P = \dfrac{1}{3}\dfrac{{mN}}{V}{v^2}\]
Here, P is the pressure in the vessel, m is the total mass of the molecules inside the vessel, N is the total number of molecules inside the vessel, V is the volume of the vessel and v is the root mean square velocity of the molecules. Thus;
\[
P\propto m{{v}^{2}} \\
\Rightarrow \dfrac{P}{{{P}_{0}}}=\dfrac{{{M}_{2}}{{v}_{2}}^{2}}{{{M}_{1}}{{v}_{1}}^{2}} \\
\Rightarrow P={{P}_{0}}\centerdot (\dfrac{1}{2})\centerdot {{(2)}^{2}} \\
\therefore P=2{{P}_{0}} \\ \]
Thus, the final pressure is twice the initial pressure.

Note: Here, we consider that the vessel is isolated from the surroundings and hence the total number of molecules and the volume of the vessel is kept constant. In addition to this, the velocity used in the formula is the root mean squared velocity which is the squared average of the velocities of all the molecules present in the vessel.