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

To solve this problem, we need to use the relationship for adiabatic processes in ideal gases, given by the equation:

\(P_1 V_1^\gamma = P_2 V_2^\gamma\)

where:

• \(P_1\) is the initial pressure.
• \(V_1\) is the initial volume.
• \(P_2\) is the final pressure.
• \(V_2\) is the final volume.
• \(\gamma\) is the adiabatic index (given here as 1.5).

We can rearrange this equation to find the ratio of initial pressure to final pressure:

\(\frac{P_1}{P_2} = \left(\frac{V_2}{V_1}\right)^\gamma\)

Substituting the given values \(V_1 = 5\, \text{litres}\) and \(V_2 = 4\, \text{litres}\):

\(\frac{P_1}{P_2} = \left(\frac{4}{5}\right)^{1.5}\)

Calculating \(\left(\frac{4}{5}\right)^{1.5}\):

\(\left(\frac{4}{5}\right)^{1.5} = \left(\frac{4}{5}\right) \times \sqrt{\left(\frac{4}{5}\right)} = \frac{4}{5} \times \frac{2}{\sqrt{5}} = \frac{8}{5\sqrt{5}}\)

Thus, the ratio of initial pressure to final pressure is:

\(\frac{8}{5\sqrt{5}}\)

Hence, the correct answer is \(\frac{8}{5\sqrt{5}}\).

Answer 2

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: \(\frac{8}{5\sqrt{5}}\).