Question

The Volume of air increases by \(5 \%\) in an adiabatic expansion. The percentage decrease in its Pressure will be (A) \(5 \%\) (B) \(6 \%\) (C) \(7 \%\) (D) \(8 \%\)

Short answer

The percentage decrease in pressure during an adiabatic expansion when the volume of air increases by 5% is approximately 7% (C).

Step-by-step solution

Express the volume increase in terms of initial and final volumes

If the volume of air increases by 5%, then the final volume can be expressed as: \(V_2 = V_1 \times (1 + 0.05) = 1.05 V_1\)

Substitute the expression for final volume into the adiabatic process equation

We can now substitute the expression for \(V_2\) into the adiabatic process equation and solve for the final pressure: \(P_1 V_1^{\gamma} = P_2 (1.05 V_1)^{\gamma}\)

Solve for the final pressure

Divide both sides of the equation by \(V_1^{\gamma}\) and use the adiabatic index value for air, \(\gamma = 1.4\): \(P_1 = P_2 (1.05)^{1.4}\) To find the final pressure, divide both sides of the equation by \((1.05)^{1.4}\): \(P_2 = \frac{P_1}{(1.05)^{1.4}}\)

Calculate the percentage decrease in pressure

The percentage decrease in pressure can be calculated as: Percentage decrease = \(\frac{P_1 - P_2}{P_1} \times 100\% = \frac{P_1 - \frac{P_1}{(1.05)^{1.4}}}{P_1} \times 100\% \) Simplify the expression: Percentage decrease = \(\frac{(1 - \frac{1}{(1.05)^{1.4}})P_1}{P_1} \times 100\% \) The \(P_1\) terms cancel out: Percentage decrease = \((1 - \frac{1}{(1.05)^{1.4}}) \times 100\% \) Calculate the percentage decrease: Percentage decrease ≈ \((1 - \frac{1}{1.0726}) \times 100\% \approx 7.26\% \)

Compare the result to the given options and identify the correct answer

The calculated percentage decrease in pressure is approximately 7.26%, which is closest to 7% among the given options. Thus, the correct answer is: (C) 7%