Question

The volume of an ideal gas (\( \gamma = 1.5 \)) is changed adiabatically from 5 litres to 4 litres. The ratio of initial pressure to final pressure is:

• A. \( \frac{4}{5} \)
• B. \( \frac{16}{25} \)
• C. \( \frac{8}{5\sqrt{5}} \)
• D. \( \frac{2}{\sqrt{5}} \)

Answer 1

The Correct Option is C

For an adiabatic process, the relation between pressure and volume is:

\[ P_iV_i^\gamma = P_fV_f^\gamma, \] where \(\gamma = 1.5\).

Substitute the given volumes (\(V_i = 5 \, \text{litres}, V_f = 4 \, \text{litres}\)):

\[ P_i (5)^{1.5} = P_f (4)^{1.5}. \]

Rearranging for \(\frac{P_i}{P_f}\):

\[ \frac{P_i}{P_f} = \frac{(4)^{1.5}}{(5)^{1.5}}. \]

Simplify:

\[ \frac{P_i}{P_f} = \left(\frac{4}{5}\right)^{1.5}. \]

Write \(1.5\) as \(\frac{3}{2}\):

\[ \frac{P_i}{P_f} = \left(\frac{4}{5}\right)^{\frac{3}{2}} = \left(\frac{4}{5}\right)^1 \times \left(\frac{4}{5}\right)^{\frac{1}{2}}. \]

Simplify each term:

\[ \frac{P_i}{P_f} = \frac{4}{5} \times \sqrt{\frac{4}{5}} = \frac{4}{5} \times \frac{2}{\sqrt{5}} = \frac{8}{5\sqrt{5}}. \]
Final Answer: \(\frac{8}{5\sqrt{5}}\).