Calculate work done by a system in an irreversible (single step) adiabatic expansion of 1 mole of a polyatomic gas (Ɣ=1.33) from 300K and pressure 10atm to 1atm.

To calculate the work done by a system during an irreversible adiabatic expansion of a polyatomic gas, we can use the equations related to the first law of thermodynamics and the properties of ideal gases. Let's break this down step by step.

Understanding the Scenario

In this situation, we have 1 mole of a polyatomic gas that expands irreversibly from an initial state characterized by a pressure of 10 atm and a temperature of 300 K to a final pressure of 1 atm. The heat capacity ratio (Ɣ) for the gas is given as 1.33. Since the process is adiabatic, we know there is no heat exchange with the surroundings.

Key Equations

The work done (W) during an irreversible expansion can be calculated using the formula:

W = -P_ext ΔV

Where:

• P_ext is the external pressure acting on the gas.
• ΔV is the change in volume.

Calculating Work Done

To find the work done, we need to determine the change in volume (ΔV). For this, we can use the ideal gas law:

PV = nRT

For the initial state (P1 = 10 atm, T1 = 300 K) and final state (P2 = 1 atm), we can calculate the respective volumes:

Initial Volume (V1)

Using the ideal gas law for the initial state:

V1 = (nRT1) / P1

Substituting the values:

• n = 1 mole
• R = 0.0821 L·atm/(K·mol) (universal gas constant)
• T1 = 300 K
• P1 = 10 atm

Calculating V1:

V1 = (1 mole * 0.0821 L·atm/(K·mol) * 300 K) / 10 atm = 2.463 L

Final Volume (V2)

Now, for the final state:

V2 = (nRT2) / P2

We need to find the final temperature (T2) using the adiabatic condition for an ideal gas:

P1 * V1^Ɣ = P2 * V2^Ɣ

To find V2 directly, we rearrange the equation but first, we know we can assume T2 decreases as the gas expands. We won't need T2 explicitly to find V2 using pressures. Rearranging gives:

V2 = V1 * (P1/P2)^(1/Ɣ) = 2.463 L * (10/1)^(1/1.33)

Calculating this gives:

V2 ≈ 2.463 L * 5.239 ≈ 12.9 L

Change in Volume (ΔV)

With the volumes calculated, we can now find ΔV:

ΔV = V2 - V1 ≈ 12.9 L - 2.463 L ≈ 10.437 L

Final Calculation of Work Done

In an irreversible expansion, we can take the external pressure (P_ext) as the final pressure (1 atm). Thus:

W = -P_ext ΔV = -1 atm * 10.437 L

To convert atm·L to joules (1 atm·L = 101.325 J):

W = -10.437 * 101.325 ≈ -1058.5 J

Conclusion

The work done by the system during the irreversible adiabatic expansion of 1 mole of the polyatomic gas is approximately -1058.5 J. The negative sign indicates that work is done by the gas on the surroundings, consistent with the expansion process.