To tackle the problem of calculating the work done on nitrogen gas during the specified processes, we need to break it down step by step. We’ll first analyze the adiabatic expansion and then the isochoric (constant volume) compression. Let's dive into the details.
Understanding the Process
We have nitrogen gas undergoing two main processes:
- Adiabatic Expansion: The gas expands without heat exchange with the surroundings.
- Isochoric Compression: The gas is compressed back to its original state at constant volume.
Step 1: Adiabatic Expansion
For an adiabatic process, we can use the following relationship involving pressure (P), volume (V), and the specific heat ratio (γ):
P1V1^γ = P2V2^γ
Where:
- P1 = initial pressure = 6 atm
- V1 = initial volume
- P2 = final pressure
- V2 = final volume = 2V1
- γ (gamma) = specific heat ratio = 1.4
Substituting the values, we get:
6 atm * V1^1.4 = P2 * (2V1)^1.4
Now, simplifying this equation:
6 = P2 * 2^1.4
Calculating 2^1.4 gives approximately 2.639. Thus:
P2 = 6 / 2.639 ≈ 2.27 atm
Step 2: Work Done During Adiabatic Expansion
The work done on the gas during an adiabatic process can be calculated using the formula:
W = (P1V1 - P2V2) / (γ - 1)
Since V2 = 2V1, we can substitute:
W = (P1V1 - P2(2V1)) / (γ - 1)
Now substituting the values:
W = (6V1 - 2.27 * 2V1) / (1.4 - 1)
This simplifies to:
W = (6V1 - 4.54V1) / 0.4
W = (1.46V1) / 0.4 = 3.65V1
Step 3: Isochoric Compression
During the isochoric process, the volume remains constant, so no work is done on the gas:
W_isochoric = 0
Calculating Total Work Done
The total work done on the gas during the entire process is simply the work done during the adiabatic expansion, as the work during the isochoric compression is zero:
Total Work Done = W_ad = 3.65V1
Visual Representation
To illustrate the processes, we can sketch a P-V diagram:
- Start at point A (P1, V1).
- Move to point B (P2, V2) during the adiabatic expansion.
- Return to point A during the isochoric compression (vertical line since volume is constant).
In the P-V diagram, the curve from A to B represents the adiabatic expansion, while the vertical line from B back to A represents the isochoric compression.
Final Thoughts
In summary, the work done on the nitrogen gas during the adiabatic expansion is expressed as 3.65V1, where V1 is the initial volume of the gas. This approach highlights the importance of understanding the thermodynamic processes involved and applying the appropriate equations to derive meaningful results.
