Question

Assuming the expression for the pressure exerted by the gas, it can be shown that pressure is

• A. \((\frac{2}{3})^{rd}\) of kinetic energy per unit volume of a gas.
• B. \((\frac{3}{4})^{th}\) of kinetic energy per unit volume of a gas.
• C. \((\frac{1}{3})^{rd}\) of kinetic energy per unit volume of a gas.
• D. \((\frac{3}{2})^{nd}\) times kinetic energy per unit volume of a gas.

Hint:

The pressure formula P = (1/3) rho v^2 can be rewritten using kinetic energy per unit volume.

Answer 1

The Correct Option is A

Step 1: Start from molecules
For $N$ molecules of mass $m$ in volume $V$: $PV = \dfrac{1}{3}Nm\overline{v^2}$.

Step 2: Total kinetic energy
The total translational kinetic energy is $K = \dfrac{1}{2}Nm\overline{v^2}$, so $Nm\overline{v^2} = 2K$.

Step 3: Relate
$PV = \dfrac{2}{3}K$, so $P = \dfrac{2}{3}\dfrac{K}{V}$.

Step 4: Answer
Pressure equals two thirds of the kinetic energy per unit volume, option (A).

Final Answer:
Pressure is two thirds of the kinetic energy per unit volume. This is option (A). \[ \boxed{\text{(A) }\frac{2}{3}} \]