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Calculate the efficiency of the reversible cycle in figure for an ideal monoatomic gas where the transformation from 1->2 is described by the equation P² V = A= constant (Assume V2 = 4 V1).

Calculate the efficiency of the reversible cycle in figure for an ideal monoatomic gas where the transformation from 1->2 is described by the equation P² V = A= constant (Assume V2 = 4 V1).

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**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.
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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