A cylinder with initial volume V contains a sample of a gas at pressure p. On one end of the cylinder, a piston is let free to move so that the gas slowly expands in such a way that its pressure is directly proportional to its volume. After the gas reaches the volume 3V and pressure 3p, the piston is pushed in so that the gas is compressed isobarically to its original volume V. The gas is then cooled isochorically until it returns to the original volume and pressure. Find the work W done on the gas during the entire process.
A cylinder with initial volume V contains a sample of a gas at pressure p. On one end of the cylinder, a piston is let free to move so that the gas slowly expands in such a way that its pressure is directly proportional to its volume. After the gas reaches the volume 3V and pressure 3p, the piston is pushed in so that the gas is compressed isobarically to its original volume V. The gas is then cooled isochorically until it returns to the original volume and pressure. Find the work W done on the gas during the entire process.
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Find the amount of work W done on the gas during the entire process?
express answer in terms of some or all of the variables p and V
W = ?
Transcribed Image Text:### Problem Statement:
A cylinder with initial volume \( V \) contains a sample of a gas at pressure \( p \). On one end of the cylinder, a piston is let free to move so that the gas slowly expands in such a way that its pressure is directly proportional to its volume. After the gas reaches the volume \( 3V \) and pressure \( 3p \), the piston is pushed in so that the gas is compressed isobarically to its original volume \( V \). The gas is then cooled isochorically until it returns to the original volume and pressure.
Find the work \( W \) done on the gas during the entire process.
### Explanation:
1. **Initial State:**
- Volume: \( V \)
- Pressure: \( p \)
2. **Expansion Process:**
- The gas expands such that the pressure is directly proportional to the volume.
- When the volume reaches \( 3V \), the pressure becomes \( 3p \).
3. **Compression Process:**
- The gas is compressed isobarically (constant pressure process) back to its original volume \( V \).
4. **Cooling Process:**
- The gas is cooled isochorically (constant volume process) until it returns to its original pressure \( p \).
### Graphical Explanation:
1. **Expansion (from \( V, p \) to \( 3V, 3p \)):**
- On a PV diagram, this process would be represented by a straight line with a positive slope originating from point \( (V, p) \) to point \( (3V, 3p) \).
2. **Isobaric Compression (from \( 3V, 3p \) to \( V, 3p \)):**
- This process appears as a horizontal line moving left from \( (3V, 3p) \) to \( (V, 3p) \).
3. **Isochoric Cooling (from \( V, 3p \) to \( V, p \)):**
- This is represented by a vertical line descending from \( (V, 3p) \) to \( (V, p) \).
### Calculation:
The work done in this cyclic process is the area enclosed by the loop on the PV diagram.
- **Work done during expansion:**
\[
W_1 = \
![---
### MODEL
Assume that the gas is ideal and the process is quasi-static.
### VISUALIZE
Show the process on a \( pV \) diagram. Note whether it happens to be one of the basic gas processes: isochoric, isobaric, or isothermal.
### SOLVE
Calculate the work as the area under the \( pV \) curve either geometrically or by carrying out the integration:
\[ \text{work done on the gas } W = -\int_{V_i}^{V_f} p \, dV = -(\text{area under } pV \text{ curve}) \]
### REVIEW
Check your signs.
- \( W > 0 \) when the gas is compressed. Energy is transferred from the environment to the gas.
- \( W < 0 \) when the gas expands. Energy is transferred from the gas to the environment.
- No work is done if the volume doesn't change, \( W = 0 \).
---
### Detailed Explanation of Diagram and Formula
**Diagram Description:**
1. **\( pV \) Diagram**: A graph where pressure \( (p) \) is plotted on the vertical axis and volume \( (V) \) is plotted on the horizontal axis. The curve or line on this graph represents the state of the gas at different volumes and pressures during the process.
2. **Process Types**:
- **Isochoric**: Vertical line (constant volume).
- **Isobaric**: Horizontal line (constant pressure).
- **Isothermal**: Curved line (constant temperature).
**Formula Explanation:**
1. **Work Done ( \( W \) )**:
- The work done on the gas during a quasi-static process is calculated by taking the negative integral of pressure \( (p) \) with respect to volume \( (V) \).
- \( W = -\int_{V_i}^{V_f} p \, dV \)
- This represents the negative area under the \( pV \) curve.
2. **Sign Convention**:
- Positive Work ( \( W > 0 \) ): Occurs when the gas is compressed (volume decreases).
- Negative Work ( \( W < 0 \) ): Occurs when the gas expands (volume increases).
- Zero Work ( \( W = 0 \) ): Occurs when there is no change in](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6ae3fb89-f8a8-4394-beb4-7c14e660d345%2Fe18495ba-98a0-4ef5-8723-667f2d0e7c45%2Foq75pr5_processed.png&w=3840&q=75)
Transcribed Image Text:---
### MODEL
Assume that the gas is ideal and the process is quasi-static.
### VISUALIZE
Show the process on a \( pV \) diagram. Note whether it happens to be one of the basic gas processes: isochoric, isobaric, or isothermal.
### SOLVE
Calculate the work as the area under the \( pV \) curve either geometrically or by carrying out the integration:
\[ \text{work done on the gas } W = -\int_{V_i}^{V_f} p \, dV = -(\text{area under } pV \text{ curve}) \]
### REVIEW
Check your signs.
- \( W > 0 \) when the gas is compressed. Energy is transferred from the environment to the gas.
- \( W < 0 \) when the gas expands. Energy is transferred from the gas to the environment.
- No work is done if the volume doesn't change, \( W = 0 \).
---
### Detailed Explanation of Diagram and Formula
**Diagram Description:**
1. **\( pV \) Diagram**: A graph where pressure \( (p) \) is plotted on the vertical axis and volume \( (V) \) is plotted on the horizontal axis. The curve or line on this graph represents the state of the gas at different volumes and pressures during the process.
2. **Process Types**:
- **Isochoric**: Vertical line (constant volume).
- **Isobaric**: Horizontal line (constant pressure).
- **Isothermal**: Curved line (constant temperature).
**Formula Explanation:**
1. **Work Done ( \( W \) )**:
- The work done on the gas during a quasi-static process is calculated by taking the negative integral of pressure \( (p) \) with respect to volume \( (V) \).
- \( W = -\int_{V_i}^{V_f} p \, dV \)
- This represents the negative area under the \( pV \) curve.
2. **Sign Convention**:
- Positive Work ( \( W > 0 \) ): Occurs when the gas is compressed (volume decreases).
- Negative Work ( \( W < 0 \) ): Occurs when the gas expands (volume increases).
- Zero Work ( \( W = 0 \) ): Occurs when there is no change in
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