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Chapter 8: Problem 1170

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

Expert verified
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

01

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: \(V_2 = V_1 \times (1 + 0.05) = 1.05 V_1\)
02

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: \(P_1 V_1^{\gamma} = P_2 (1.05 V_1)^{\gamma}\)
03

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 \((1.05)^{1.4}\): \(P_2 = \frac{P_1}{(1.05)^{1.4}}\)
04

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 ≈ \((1 - \frac{1}{1.0726}) \times 100\% \approx 7.26\% \)
05

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%

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Most popular questions from this chapter

A monoatomic ideal gas, intially at temperature \(1_{1}\) is enclosed in a cylinders fitted with a frictionless piston. The gas is allowed to expand adiabatically to a temperature \(\mathrm{T}_{2}\) by releasing the piston suddenly If \(\mathrm{L}_{1}\) and \(\mathrm{L}_{2}\) the lengths of the gas column before and after expansion respectively, then then \(\left(\mathrm{T}_{1} / \mathrm{T}_{2}\right)\) is given by (A) \(\left\\{\mathrm{L}_{1} / \mathrm{L}_{2}\right\\}^{(2 / 3)}\) (B) \(\left\\{\mathrm{L}_{2} / \mathrm{L}_{1}\right\\}^{(2 / 3)}\) (C) \(\left\\{\mathrm{L}_{1} / \mathrm{L}_{2}\right\\}\) (D) \(\left\\{\mathrm{L}_{2} / \mathrm{L}_{1}\right\\}\)

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In a thermodynamic process, pressure of a fixed mass of a gas is changed in such a manner that the gas release \(20 \mathrm{~J}\) of heat and $8 \mathrm{~J}$ of work has done on the gas-If the initial internal energy of the gas was \(30 \mathrm{j}\), then the final internal energy will be (A) \(58 \mathrm{~J}\) (B) \(2 \mathrm{~J}\) (C) \(42 \mathrm{~J}\) (D) \(18 \mathrm{~J}\)

Instructions:Read the assertion and reason carefully to mask the correct option out of the options given below. (A) If both assertion and reason are true and the reason is the correct explanation of the assertion. (B) If both assertion and reason are true but reason is not be correct explanation of assertion. (C) If assertion is true but reason is false. (D) If the assertion and reason both are false. Assertion: The carnot is useful in understanding the performance of heat engine Reason: The carnot cycle provides a way of determining the maximum possible efficiency achievable with reservoirs of given temperatures. (A) \(\mathrm{A}\) (B) B (C) \(\mathrm{C}\) (D) \(\mathrm{D}\)
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