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How are equations 1 and 2 combined to that? Especially, where does the (n-1)/n comes from

How are equations 1 and 2 combined to that? Especially, where does the (n-1)/n comes from

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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How are equations 1 and 2 combined to that? Especially, where does the (n-1)/n comes from

**Polytropic Expansion Process** **Expression of Polytropic Expansion Process:** \[ PV^n = \text{Constant} \quad \text{(I)} \] - **Variables:** - \( P \) is the pressure - \( V \) is the volume - \( n \) is the polytropic index **Expression of Ideal Gas Process:** \[ PV = RT \quad \text{(II)} \] - **Variables:** - \( T \) is the temperature - \( R \) is the gas constant **Combining Equations (I) and (II):** \[ T_2 = T_1 \left( \frac{P_1}{P_2} \right)^{(n-1)/n} \quad \text{(III)} \] - **Variables:** - \( P_1 \) is the initial absolute pressure - \( P_2 \) is the final absolute pressure - \( T_1 \) is the initial temperature **Conclusion:** Substitute \( 1200 \, \text{kPa} \) for \( P_1 \), \( 120 \, \text{kPa} \) for \( P_2 \), and \( 303 \, \text{K} \) for \( T_1 \) in Equation (III). \[ T_2 = (303 \, \text{K}) \left( \frac{1200 \, \text{kPa}}{120 \, \text{kPa}} \right)^{0.2/1.2} \] \[ T_2 = 445 \, \text{K} \] Thus, the final temperature in a polytropic process is **445 K**.
Transcribed Image Text:**Polytropic Expansion Process** **Expression of Polytropic Expansion Process:** \[ PV^n = \text{Constant} \quad \text{(I)} \] - **Variables:** - \( P \) is the pressure - \( V \) is the volume - \( n \) is the polytropic index **Expression of Ideal Gas Process:** \[ PV = RT \quad \text{(II)} \] - **Variables:** - \( T \) is the temperature - \( R \) is the gas constant **Combining Equations (I) and (II):** \[ T_2 = T_1 \left( \frac{P_1}{P_2} \right)^{(n-1)/n} \quad \text{(III)} \] - **Variables:** - \( P_1 \) is the initial absolute pressure - \( P_2 \) is the final absolute pressure - \( T_1 \) is the initial temperature **Conclusion:** Substitute \( 1200 \, \text{kPa} \) for \( P_1 \), \( 120 \, \text{kPa} \) for \( P_2 \), and \( 303 \, \text{K} \) for \( T_1 \) in Equation (III). \[ T_2 = (303 \, \text{K}) \left( \frac{1200 \, \text{kPa}}{120 \, \text{kPa}} \right)^{0.2/1.2} \] \[ T_2 = 445 \, \text{K} \] Thus, the final temperature in a polytropic process is **445 K**.
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