Question

Consider a given sample of an ideal gas (C_p/C_v = 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.

Answer 1

(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 = ?`

P₂V₂^γ = P₃V₃^γ

`=> ("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) P₁V₁^γ = P₂V₂^γ

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`