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![**Title: Calculating the Efficiency of a Reversible Cycle for an Ideal Monoatomic Gas**
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**Objective:**
Our goal is to calculate the efficiency of the reversible cycle depicted in the figure for an ideal monoatomic gas, where the transformation from state 1 to state 2 follows the given equation \( P^2 V = A = \text{constant} \). Assume \( V_2 = 4 V_1 \).
**Graph Description:**
- The vertical axis is labeled \( P \) (pressure).
- The horizontal axis is labeled \( V \) (volume).
- Point 1 is at a higher pressure and lower volume.
- Point 2 is at a lower pressure and higher volume, with \( V_2 = 4 V_1 \).
- The path from point 1 to point 2 follows a curve that represents the equation \( P^2 V = A \), indicating an inversely proportional relationship between \( P \) and \( V \).
- Point 3 coordinates with \( P_1 \) and \( V_1 \), and moves horizontally to the right for \( V_2 \).
**Hint:**
You may need to use the following integral:
\[ \int \frac{dx}{\sqrt{x}} = 2 \sqrt{x} + C \]
Additionally, for the transition from state 1 to state 2, the expression:
\[ Q_{1 \rightarrow 2} = \Delta U + W \]
may be useful, where the integral provided might come in handy.
**Steps to Calculating Efficiency:**
1. Understand the given equation and the graph.
2. Use the relationship \( P^2 V = A \) to analyze changes in pressure and volume.
3. Apply the integral \( \int \frac{dx}{\sqrt{x}} = 2 \sqrt{x} + C \) if required.
4. Utilize the thermodynamic relationship \( Q_{1 \rightarrow 2} = \Delta U + W \) where applicable.
**Conclusion:**
By evaluating the gas's transitions and using the relevant integral and thermodynamic relationship, we can derive the efficiency of the specified reversible cycle for an ideal monoatomic gas.
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This content is aimed at guiding students through the process of calculating the efficiency of a reversible cycle involving an ideal monoatomic gas, using both graphical analysis and thermodynamic principles.](https://content.bartleby.com/qna-images/question/d2310cfd-2850-4e63-bb0e-d2d4ff66fc6a/c967f895-b84b-4f92-8434-d7ce3877c244/6kywuo_processed.jpeg)
Transcribed Image Text:**Title: Calculating the Efficiency of a Reversible Cycle for an Ideal Monoatomic Gas**
---
**Objective:**
Our goal is to calculate the efficiency of the reversible cycle depicted in the figure for an ideal monoatomic gas, where the transformation from state 1 to state 2 follows the given equation \( P^2 V = A = \text{constant} \). Assume \( V_2 = 4 V_1 \).
**Graph Description:**
- The vertical axis is labeled \( P \) (pressure).
- The horizontal axis is labeled \( V \) (volume).
- Point 1 is at a higher pressure and lower volume.
- Point 2 is at a lower pressure and higher volume, with \( V_2 = 4 V_1 \).
- The path from point 1 to point 2 follows a curve that represents the equation \( P^2 V = A \), indicating an inversely proportional relationship between \( P \) and \( V \).
- Point 3 coordinates with \( P_1 \) and \( V_1 \), and moves horizontally to the right for \( V_2 \).
**Hint:**
You may need to use the following integral:
\[ \int \frac{dx}{\sqrt{x}} = 2 \sqrt{x} + C \]
Additionally, for the transition from state 1 to state 2, the expression:
\[ Q_{1 \rightarrow 2} = \Delta U + W \]
may be useful, where the integral provided might come in handy.
**Steps to Calculating Efficiency:**
1. Understand the given equation and the graph.
2. Use the relationship \( P^2 V = A \) to analyze changes in pressure and volume.
3. Apply the integral \( \int \frac{dx}{\sqrt{x}} = 2 \sqrt{x} + C \) if required.
4. Utilize the thermodynamic relationship \( Q_{1 \rightarrow 2} = \Delta U + W \) where applicable.
**Conclusion:**
By evaluating the gas's transitions and using the relevant integral and thermodynamic relationship, we can derive the efficiency of the specified reversible cycle for an ideal monoatomic gas.
---
This content is aimed at guiding students through the process of calculating the efficiency of a reversible cycle involving an ideal monoatomic gas, using both graphical analysis and thermodynamic principles.
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