To tackle this problem, we need to analyze the thermodynamic processes involved with the ideal gas and calculate the work done and heat exchanged during each step. Let’s break it down step by step, starting with the sketches for the P-V and P-T diagrams, followed by the calculations for work and heat.
Understanding the Process Steps
The process consists of three main steps:
- Isothermal Expansion: The gas expands isothermally to twice its initial volume.
- Isobaric Compression: The gas is then compressed at constant pressure back to its original volume.
- Isochoric Heating: Finally, the gas is heated at constant volume to return to its original pressure.
P-V Diagram Sketch
In the P-V diagram:
- Start at point A (V1, P). The gas expands isothermally to point B (2V1, P), moving horizontally to the right.
- From point B, the gas is compressed at constant pressure to point C (V1, P), moving vertically downwards.
- Finally, the gas is heated at constant volume from point C to point D (V1, P2), moving vertically upwards.
P-T Diagram Sketch
In the P-T diagram:
- Start at point A (T, P). The temperature remains constant during the isothermal expansion, so it stays at T while moving to point B.
- During the isobaric compression, the temperature decreases as the gas returns to its original volume, moving from point B to point C.
- During the isochoric heating, the temperature increases back to its original pressure, moving from point C to point D.
Calculating Work Done by the Gas
Let's calculate the work done during each step:
1. Isothermal Expansion
The work done during an isothermal process can be calculated using the formula:
W = nRT ln(Vf/Vi)
Here, n = 3 moles, R is the ideal gas constant, T is the temperature, Vf = 2V1, and Vi = V1.
Thus, the work done during the isothermal expansion is:
W1 = 3RT ln(2)
2. Isobaric Compression
For the isobaric process, the work done is given by:
W = P(Vf - Vi)
Here, Vf = V1 and Vi = 2V1, so:
W2 = P(V1 - 2V1) = -PV1
3. Isochoric Heating
In an isochoric process, no work is done:
W3 = 0
Total Work Done
The total work done by the gas during the entire process is:
W_total = W1 + W2 + W3 = 3RT ln(2) - PV1 + 0
Calculating Net Heat Supplied to the Gas
Now, let’s calculate the heat exchanged during each step:
1. Isothermal Expansion
The heat added during the isothermal expansion is equal to the work done:
Q1 = W1 = 3RT ln(2)
2. Isobaric Compression
The heat exchanged during the isobaric process can be calculated using:
Q = nCpΔT
Since Cp = 7R/2, and the temperature change can be found from the ideal gas law:
ΔT = (P * V1) / (nR) - (P * 2V1) / (nR) = -P * V1 / (3R)
Thus, the heat exchanged is:
Q2 = nCpΔT = 3 * (7R/2) * (-P * V1 / (3R)) = -7PV1/2
3. Isochoric Heating
For the isochoric process, the heat added is:
Q3 = nCvΔT
Where Cv = Cp - R = (7R/2) - R = 5R/2. The temperature change is from T to T2, where T2 can be calculated from the ideal gas law:
ΔT = (P2 * V1) / (nR) - T
However, since we are returning to the original pressure, we can assume the heat added is:
Q3 = nCv(T2 - T) = nCv(ΔT) = 3 * (5R/2) * (T2 - T)
But since T2 is equal to T, this simplifies to zero.
Total Heat Supplied
The total heat supplied to the gas during the entire process is:
Q_total = Q1 + Q2 + Q3 = 3RT ln(2) - 7PV1/2 + 0
Final Summary
In summary, we have:
- Net Work Done: W_total = 3RT ln(2) - PV1
- Net Heat Supplied: Q_total = 3RT ln(2) - 7PV1/2
This analysis provides a comprehensive understanding of the thermodynamic processes involved, along with the calculations for work and heat. If you have any further questions or need clarification on any specific part, feel free to ask!
