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Question: An ideal gas is compressed isothermally until its pressure is doubled and then allowed to expand adi...

An ideal gas is compressed isothermally until its pressure is doubled and then allowed to expand adiabatically to regain its original volume (g = 1.4 and 2–1.4 = 0.38). The ratio of the final to initial pressure is

A

0.76 : 1

B

1 : 1

C

0.66 : 1

D

0.86 : 1

Answer

0.76 : 1

Explanation

Solution

Let V be the original volume of the gas. For a isothermal process PV = constant

PiTi=PfVf\therefore P_{i}T_{i} = P_{f}V_{f}

Vf=Vi(PiPf)=V(Pi2Pi()V2)V_{f} = V_{i}\left( \frac{P_{i}}{P_{f}} \right) = V\left( \frac{}{P_{i}{2P_{i}}}()\frac{V}{2} \right)

For an adiabatic process

PVγPV^{\gamma}= Constant

According to questions

(2Pi)(V2)γ=PfVγ\therefore(2P_{i})\left( \frac{V}{2} \right)^{\gamma} = P_{f}V^{\gamma}

PfPi=2(2)γ=2(2)1,4=2(0.38)=0.761\frac{P_{f}}{P_{i}} = 2(2)^{- \gamma} = 2(2)^{- 1,4} = 2(0.38) = \frac{0.76}{1}