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

The problem is solved using the adiabatic process relationship for ideal gases, described by the equation:

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

Variables are defined as:

• \(P_1\): Initial pressure.
• \(V_1\): Initial volume.
• \(P_2\): Final pressure.
• \(V_2\): Final volume.
• \(\gamma\): Adiabatic index (1.5).

Rearranging the equation to determine the initial to final pressure ratio yields:

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

With the given volumes \(V_1 = 5\, \text{litres}\) and \(V_2 = 4\, \text{litres}\), the equation becomes:

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

The calculation of \(\left(\frac{4}{5}\right)^{1.5}\) is as follows:

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

Therefore, the pressure ratio is:

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

The final answer is \(\frac{8}{5\sqrt{5}}\).