A mass of ideal gas of volume 400 cm3 at a temperature of 27 ˚C expands adiabatically
until its volume is 500 cm3. Calculate the new temperature. The gas is then compressed
isothermally until its pressure returns to the original value. Calculate the final volume of
the gas. Assume γ = 1.40.
To solve this problem, we will use the adiabatic and isothermal processes equations for ideal gases.
1. Adiabatic process:
For an adiabatic process, we have the equation:
PV^(γ) = constant,
where P is the pressure, V is the volume, and γ (gamma) is the heat capacity ratio for the gas.
Given:
Initial volume, V1 = 400 cm^3
Initial temperature, T1 = 27 ˚C = 27 + 273.15 K
Final volume, V2 = 500 cm^3
Heat capacity ratio, γ = 1.40
First, we need to convert the initial temperature from Celsius to Kelvin:
T1 = 27 + 273.15 = 300.15 K
Using the equation of state for adiabatic expansion, we can write:
P1 * V1^(γ) = P2 * V2^(γ),
where P1 and P2 are the initial and final pressures, respectively.
Since the process is adiabatic, we have P1 = P2.
So, rearranging the equation, we get:
V1^(γ) / V2^(γ) = V1 / V2.
Plugging in the given values, we get:
(400 cm^3 / 500 cm^3)^(1.40) = V1 / V2.
Simplifying, we find:
(0.8)^(1.40) = V1 / V2.
Now, solve for V2:
V2 = V1 / (0.8)^(1.40).
Substituting the known values, we get:
V2 = 400 cm^3 / (0.8)^(1.40).
Calculating the result using a calculator, we find:
V2 ≈ 480.88 cm^3.
Therefore, the final volume of the gas after the adiabatic expansion is approximately 480.88 cm^3.
2. Isothermal process:
For an isothermal process, the equation relates pressure and volume:
P1 * V1 = P2 * V2.
Given:
Initial pressure, P1 = P2 (as mentioned before),
Initial volume, V1 = 400 cm^3 (as mentioned before).
Final volume, V2 = 480.88 cm^3 (obtained from the adiabatic process),
Simplifying the equation, we find:
V2 = (P1 * V1) / P2.
Since P1 = P2 (isothermal compression at the same pressure), we can rewrite the equation as:
V2 = V1.
So, the final volume of the gas after the isothermal compression is equal to the initial volume:
V2 = 400 cm^3.
Therefore, the final volume of the gas after the isothermal compression is 400 cm^3.