Initially 1.200 mol of an ideal gas in a container occupies a volume of 3.50 l at a pressure of 3.30 atm with an internal energy U1 = 547.2 J. The gas is cooled at a constant volume until its pressure is 1.80 atm. Then it is allowed to expand at constant pressure until its volume is 7.90 l. The final internal energy is U2 = 673.7 J. All processes are quasi static. What is the work done by the gas?

Question

Initially 1.200 mol of an ideal gas in a container occupies a volume of 3.50 l at a pressure of 3.30 atm with an internal energy U1 = 547.2 J. The gas is cooled at a constant volume until its pressure is 1.80 atm. Then it is allowed to expand at constant pressure until its volume is 7.90 l. The final internal energy is U2 = 673.7 J. All processes are quasi static. What is the work done by the gas?

Answer 1

To determine the work done by the gas, we need to calculate the change in internal energy (ΔU) and then use the first law of thermodynamics.

The first law of thermodynamics states that the change in internal energy (ΔU) of a system is equal to the heat (Q) added to the system minus the work (W) done by the system:

ΔU = Q - W

In this case, the gas undergoes two processes:

1. Constant volume process: Here, the volume remains constant, so no work is done (W = 0). Therefore, the change in internal energy (ΔU1) is simply equal to the heat added (Q1):
ΔU1 = Q1

2. Constant pressure process: Here, the pressure remains constant. The work done (W2) can be calculated using the formula:
W2 = PΔV

Now, let's calculate the values step by step.

1. Constant volume process:
Since the volume remains constant (V1 = V2), the work done is zero (W1 = 0). Therefore, ΔU1 = Q1 = 547.2 J.

2. Constant pressure process:
We need to calculate the work done using the formula W2 = PΔV. Here, the initial pressure (P1) is 3.30 atm, the final pressure (P2) is 1.80 atm, and the change in volume (ΔV) is V2 - V1.

ΔV = V2 - V1
ΔV = 7.90 L - 3.50 L
ΔV = 4.40 L

Now, substitute the values into the formula:
W2 = PΔV
W2 = 1.80 atm * 4.40 L

To convert atm to Joules, we need to multiply by the conversion factor 101.325 J/atm:
W2 = (1.80 atm * 101.325 J/atm) * 4.40 L

Finally, simplify the equation:
W2 = 727.65 J * 4.40
W2 = 3201.96 J

Now that we have calculated the work done during the constant pressure process, we can calculate the total work done by adding the work done in both processes:

W = W1 + W2
W = 0 + 3201.96 J
W = 3201.96 J

Therefore, the work done by the gas is 3201.96 Joules.