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: Recall the formula
From kinetic theory, \(P = \dfrac{1}{3}\rho\,\overline{v^2}\) where \(\rho\) is the density of the gas and \(\overline{v^2}\) is the mean square speed.

Step 2: Rewrite using energy
The translational kinetic energy per unit volume is \(E = \dfrac{1}{2}\rho\,\overline{v^2}\), so \(\rho\,\overline{v^2} = 2E\).

Step 3: Substitute
\[ P = \frac{1}{3}\times2E = \frac{2}{3}E \]

Step 4: Result
The pressure is two thirds of the kinetic energy per unit volume, option (A). The fractions \(\frac{1}{3}\) and \(\frac{3}{4}\) and \(\frac{3}{2}\) do not result from this substitution.

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