Question

The average speed of gas molecules is v at pressure p if by keeping temperature constant the pressure of gas is doubled then average speed will become?

Answer 1

To solve this question, we need to know the mathematical formula for average speed of gas molecules and Ideal gas equation, which is (for n mole): $PV=nRT$
In the calculation process use the ideal gas equation for 1 mole gas, and put the newly found values for pressure (P) and volume (V) into the average speed of the gas molecules formula.

Formula used:
$v=\sqrt{\dfrac{RT}{m{{N}_{A}}\pi }}$

Complete step-by-step answer:
Average speed of gas molecules is given by,
$v=\sqrt{\dfrac{RT}{m{{N}_{A}}\pi }}$
Where,
P is the pressure of the gas
V is the volume of gas
n is the mole
R is the gas constant
T is the temperature
${{N}_{A}}$ is the Avogadro number.
For 1 mole gas, ideal gas equation will be
$PV=RT..........(1)$
By using above formula, we can write the average speed of 1 mole gas molecules as;
${{v}_{1}}=\sqrt{\dfrac{{{P}_{1}}{{V}_{1}}}{m{{N}_{A}}\pi }}$
Now, we are given that at constant temperature pressure is doubled, then we can find the new volume for gas as;
For constant temperature: ${{P}_{1}}{{V}_{1}}={{P}_{2}}{{V}_{2}}............(2)$
Since, RT is constant now in equation (1).
Putting values of new pressure and volume into the speed formula, we get;
${{v}_{2}}=\sqrt{\dfrac{{{P}_{2}}{{V}_{2}}}{m{{N}_{A}}\pi }}$
From equation (2), we get
${{v}_{2}}=\sqrt{\dfrac{{{P}_{1}}{{V}_{1}}}{m{{N}_{A}}\pi }}={{v}_{1}}$
Thus, we again get the same velocity value.
So, ${{v}_{2}}={{v}_{1}}$
Therefore, when the average speed of gas molecules is v at pressure p if by keeping temperature constant the pressure of gas is doubled then average speed will become equal.

Note: Gaseous molecule’s motion is governed by the Kinetic Molecular Theory model;
(1) Gas molecules are in constant motion and exhibit perfectly elastic collisions.
(2) Kinetic Molecular Theory can be used to explain both Charles’s law and Boyle’s law
(3) The average kinetic energy of gas molecules is directly proportional to the temperature (T).