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Question
Consider a given sample of an ideal gas (Cp/Cv = γ) having initial pressure p0 and volume V0. (a) The gas is isothermally taken to a pressure p0/2 and from there, adiabatically to a pressure p0/4. Find the final volume. (b) The gas is brought back to its initial state. It is adiabatically taken to a pressure p0/2 and from there, isothermally to a pressure p0/4. Find the final volume.
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Solution
(a) Given,
Initial pressure of the gas = p0
Initial volume of the gas =V0
For an isothermal process,
PV = constant
So, P1V1 = P2V2
`"P"_2 = ("P"_0"V"_0)/("P"_0/2) = 2"V"_0`
For an adiabatic process , `"P"_3 = "P"_0/4 , "V"_3 = ?`
P2V2γ = P3V3γ
`=> ("V"_3/"V"_2)^gamma = ("P"_2/"P"_3)`
`=> ("V"_3/"V"_2)^ gamma = (("P"_0/2)/("P"_0 /4))= 2`
`=> "V"_3/"V"_2 = 2^(1/ gamma)`
`therefore "V"_3 = "V"_2 2^ (1/gamma) = 2"V"_0 2^(1/gamma)`
`= 2 ^[(γ +1)/γ] "V"_0`
(b) P1V1γ = P2V2γ
Or `("V"_2/"V"_1) = ("P"_1/"P"_2)^(1/gamma)`
`=> "V"_2 = "V"_0 2^(1/gamma)`
Again, for an isothermal process,
P2V2 = P3V3
`therefore "V"_3 = ("P"_2"V"_2)/"P"_3 = 22^(1/gamma)"V"_0`
`= 2 ^[(γ +1)/γ] "V"_0`
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