10 litres of a monoatomic ideal gas at 0degree celsius and 10atm pressure is suddenly released to 1atm and the gas expands adiabatically against this constant pressure. find the final temperature and volume

To find the final temperature and volume of the gas after expanding adiabatically against a constant pressure, we can use the ideal gas law and the adiabatic process equation.

1. Start with the ideal gas law:

PV = nRT

Where P is the pressure, V is the volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature in Kelvin.

2. Rearrange the ideal gas law to solve for n, the number of moles of gas:

n = PV / RT

3. Since the gas is monoatomic, we can use the equation:

Cv = (3/2)R

Where Cv is the molar specific heat capacity at constant volume.

4. Use the adiabatic process equation:

T2 / T1 = (V1 / V2)^(γ-1)

Where T1 is the initial temperature, T2 is the final temperature, V1 is the initial volume, V2 is the final volume, and γ is the heat capacity ratio (also known as the ratio of specific heats).

5. Substitute the equations from steps 2 and 3 into the adiabatic process equation:

(PV1 / nR) / (PV2 / nR) = (V1 / V2)^(γ-1)

Cancel out the factors of n and R:

(PV1) / (PV2) = (V1 / V2)^(γ-1)

6. Simplify the equation:

V1 / V2 = (PV1 / PV2)^(1/(γ-1))

Substitute the pressure values:

V1 / V2 = (P1 / P2)^(1/(γ-1))

7. Substitute the values into the equation:

V1 / V2 = (10atm / 1atm)^(1/(γ-1)) = 10^(1/(γ-1))
γ = Cp / Cv = Cp / ((3/2)R)

8. The value of γ for a monoatomic ideal gas is approximately 5/3, so substitute the value of γ into the equation:

V1 / V2 = 10^(1/(5/3-1)) = 10^(3/2)

9. Solve for V2, the final volume:

V2 = V1 / 10^(3/2)

10. Calculate V2 using the given initial volume:

V2 = 10 L / 10^(3/2) = 10 L / (√10) = 10 / √10 L ≈ 3.16 L

11. To find the final temperature, use the adiabatic process equation:

T2 / T1 = (V1 / V2)^(γ-1)

Substitute in the known values:

T2 / 273K = (10 L / 3.16 L)^(5/3-1)

Now solve for T2:

T2 = 273K * ((10 L / 3.16 L)^(5/3-1))

Calculate T2 using the given initial temperature:

T2 ≈ 273K * (3.16^(2/3)) ≈ 273K * 1.791 ≈ 489.543K

Therefore, the final temperature of the gas is approximately 489.543K and the final volume is approximately 3.16 L.