Let the initial volume be V₁ and initial pressure be P₁ = 1 atm. After compressing adiabatically let the new volume be V₂ and the new pressure is P₂ = 5 atm. From there it is isothermally expanded to the initial volume V₁ and the pressure now is P₃.

• Adiabatic Compression:
α = Adiabatic constant = 1.4
P₁(V₁)ᵅ= P₂(V₂)ᵅ
(V₁/V₂)ᵅ = P₂/P₁ = 5
⇒V₁/V₂ = 5⁰·⁷¹
⇒V₁/V₂ = 3.15

• Isothermal Expansion:
P₂V₂ = P₃V₁
P₃ = P₂ × (V₂/V₁)
= 5/3.15
= 1.58 atm

Therefore the final pressure at the end of Isothermal process is 1.58 atm.

Adiabatic Compression and Isothermal Expansion of Gas

Adiabatic Compression:
When a gas is compressed adiabatically, its temperature increases and its pressure also increases. The adiabatic compression is a process in which no heat is added or removed from the system. The equation for adiabatic process is PV^γ=constant, where P is the pressure, V is the volume, and γ is the ratio of specific heats.

Isothermal Expansion:
When a gas is allowed to expand isothermally, its temperature remains constant and its pressure decreases. The isothermal expansion is a process in which the temperature of the system remains constant. The equation for isothermal process is PV= constant, where P is the pressure, V is the volume, and T is the temperature.

Process at the End of the Isothermal Process:
In this question, a fixed mass of gas (Y=1.4) at 1 atm is compressed adiabatically to 5 atm. Using the equation for adiabatic process, we can find the final volume of the gas after adiabatic compression. Let's assume the initial volume of the gas to be V1 and the final volume to be V2.

PV^γ=constant
P1V1^γ=P2V2^γ
V2=V1( P1/P2)^1/γ
V2=V1(1/5)^1/1.4

After adiabatic compression, the gas is allowed to expand isothermally to its original volume. Using the equation for isothermal process, we can find the final pressure of the gas after isothermal expansion. Let's assume the final pressure of the gas to be P3.

PV=constant
P2V2=P3V1
P3=P2V2/V1
P3=5(1/5)^1/1.4

Therefore, the final pressure of the gas after isothermal expansion is 1 atm.