Air expands in a piston-cylinder from 500 kPa to 100 kPa in a polytropic process with n 1.5. The initial temperature and volume of the air are 400 K and 1 m³, respectively. Treat the air as an ideal gas with constant properties at 400 K. How much work is produced by the air during this expansion?
Air expands in a piston-cylinder from 500 kPa to 100 kPa in a polytropic process with n 1.5. The initial temperature and volume of the air are 400 K and 1 m³, respectively. Treat the air as an ideal gas with constant properties at 400 K. How much work is produced by the air during this expansion?
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![**Problem Statement:**
Air expands in a piston-cylinder from 500 kPa to 100 kPa in a polytropic process with \( n = 1.5 \). The initial temperature and volume of the air are 400 K and 1 m\(^3\), respectively. Treat the air as an ideal gas with constant properties at 400 K.
**Question:**
How much work is produced by the air during this expansion?
**Explanation:**
This problem involves calculating the work done during a polytropic process, which is a type of thermodynamic process that follows the relation \( PV^n = \text{constant} \), where \( P \) is pressure, \( V \) is volume, and \( n \) is the polytropic index.
The work done in a polytropic process can be calculated using the formula:
\[ W = \frac{P_2 V_2 - P_1 V_1}{n - 1} \]
Where:
- \( P_1 \) and \( P_2 \) are the initial and final pressures,
- \( V_1 \) and \( V_2 \) are the initial and final volumes,
- \( n \) is the polytropic index.
Given the ideal gas nature, additional relations can be derived using the ideal gas law to find final volume or other unknowns if required.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb78840cf-991a-43cf-baf3-7d430789620d%2F14f7181f-1c4f-41c4-afcb-883a5cccb9d8%2Fv4tabjc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Air expands in a piston-cylinder from 500 kPa to 100 kPa in a polytropic process with \( n = 1.5 \). The initial temperature and volume of the air are 400 K and 1 m\(^3\), respectively. Treat the air as an ideal gas with constant properties at 400 K.
**Question:**
How much work is produced by the air during this expansion?
**Explanation:**
This problem involves calculating the work done during a polytropic process, which is a type of thermodynamic process that follows the relation \( PV^n = \text{constant} \), where \( P \) is pressure, \( V \) is volume, and \( n \) is the polytropic index.
The work done in a polytropic process can be calculated using the formula:
\[ W = \frac{P_2 V_2 - P_1 V_1}{n - 1} \]
Where:
- \( P_1 \) and \( P_2 \) are the initial and final pressures,
- \( V_1 \) and \( V_2 \) are the initial and final volumes,
- \( n \) is the polytropic index.
Given the ideal gas nature, additional relations can be derived using the ideal gas law to find final volume or other unknowns if required.
Expert Solution
Step 1
Given:
P1=500kPa
T1=400k
V1= 1m3
P2=100kPa
To find:
Amount of work is produced by the air during expansion.
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