Question

In an ideal gas at temperature T, the average force that a molecule applies on the walls of a closed container depends on T as $ {{T}^{q}} $. What would be the good estimate of q?
(A) 1/4
(B) 1/2
(C) 1
(D) 2

Answer 1

Hint: Use the formula for the ideal gas and the pressure formula in terms of mass and volume for the ideal gas to get a relation between the root mean square (r.m.s.) velocity and the temperature and then relate the temperature of the ideal gas to the force using the rate of change of momentum formula i.e. using Newton’s second law of motion.

Complete step by step answer:
As we know that the pressure of the ideal gas is given by:
 $ \begin{align}
 &\Rightarrow P=\dfrac{1}{3}\dfrac{mN}{V}{{v}_{rms}}^{2}\text{ }.......\text{(1)} \\
 & \text{where m is the mass of the molecule} \\
 & N\text{ is the number of molecules per unit V volume} \\
 & {{v}_{rms}}\text{ is the root mean square velocity of gas molecules} \\
 & Also \\
 &\Rightarrow PV=\dfrac{mN}{V}T\ \text{ }.........\text{(2)} \\
\end{align} $
On comparing the equations (1) and (2) we get,
 $ {{v}_{rms}}^{2}\text{ }\alpha \text{ }T $ (If mass of the gas molecules and its temperature are constant)
Also for force;
 $ \begin{align}
 &\Rightarrow F=\dfrac{dp}{dt}\text{ (rate of change of momentum)} \\
 &\Rightarrow \text{=}\dfrac{m{{v}_{rms}}^{2}}{L}\text{ } \\
 &\Rightarrow \text{where L is the length of the container} \\
\end{align} $
We Note: that,
 $ \begin{align}
 &\Rightarrow F\text{ }\alpha \text{ }{{v}_{rms}}^{2}\text{ }\alpha \text{ }T \\
 & So, \\
 &\Rightarrow F\text{ }\alpha \text{ }T \\
\end{align} $
Therefore the estimated value of q is 1.

Note:
You can also relate the pressure and temperature from the above formulas and as $ P\text{ }\alpha \text{ }F $. we can directly find the relation between force and temperature and hence the value of q without using the mean square velocity as medium for relation can be estimated.