A certain ideal gas has the following properties: Cpo = A + (B/T) where A and B are constants and T is temperature in °K and Cpo has units kJ / K /kg. The gas constant is R = 0.25 kJ/ºK/kg and the constants A = 1.25, B = 173.8029748. For T₁ = 300 °K and T₂ = 400 °K, determine the change in specific internal energy, (u₂-u₁ ) in kJ/kg. You may NOT assume constant specific heats.
A certain ideal gas has the following properties: Cpo = A + (B/T) where A and B are constants and T is temperature in °K and Cpo has units kJ / K /kg. The gas constant is R = 0.25 kJ/ºK/kg and the constants A = 1.25, B = 173.8029748. For T₁ = 300 °K and T₂ = 400 °K, determine the change in specific internal energy, (u₂-u₁ ) in kJ/kg. You may NOT assume constant specific heats.
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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![### Ideal Gas Properties and Specific Internal Energy Calculation
In this section, we will explore the properties of a specific ideal gas and determine the change in specific internal energy given varying temperature conditions.
#### Given Properties and Formula
A particular ideal gas is characterized by the following equation for heat capacity at constant pressure \( C_{p0} \):
\[ C_{p0} = A + \left( \frac{B}{T} \right) \]
where:
- \( A \) and \( B \) are constants.
- \( T \) is the temperature in Kelvin (\( °K \)).
- \( C_{p0} \) has units of \( \text{kJ} / °K \cdot \text{kg} \).
The gas constant for this system is given by:
\[ R = 0.25 \text{kJ} / °K \cdot \text{kg} \]
The constants are:
\[ A = 1.25 \]
\[ B = 173.8029748 \]
We are given two specific temperatures:
- \( T_1 = 300°K \)
- \( T_2 = 400°K \)
#### Objective
Determine the change in specific internal energy (\( u_2 - u_1 \)) in \( \text{kJ} / \text{kg} \). It is important to note that the assumption of constant specific heats is not valid for this calculation.
#### Process
1. **Integrate the specific heat capacity function to determine the change in internal energy:**
Specific heat capacity at constant volume \( C_v \) can be related to \( C_{p0} \) as follows, considering the ideal gas relationship:
\[ C_v = C_{p0} - R \]
2. **Integrate \( C_v \) over the temperature range:**
\[ u_2 - u_1 = \int_{T_1}^{T_2} C_v \, dT \]
\[ \Rightarrow u_2 - u_1 = \int_{T_1}^{T_2} \left( A + \frac{B}{T} - R \right) \, dT \]
Substitute the values of \(A\), \(B\), and \(R\):
\[ u_2 - u_1 = \int_{T_1}^{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe3acf6a6-30f5-4bc7-ac16-270edca78ecc%2F02921b71-6598-41a9-8aa3-b91c4eab5786%2Fun9rqn_processed.png&w=3840&q=75)
Transcribed Image Text:### Ideal Gas Properties and Specific Internal Energy Calculation
In this section, we will explore the properties of a specific ideal gas and determine the change in specific internal energy given varying temperature conditions.
#### Given Properties and Formula
A particular ideal gas is characterized by the following equation for heat capacity at constant pressure \( C_{p0} \):
\[ C_{p0} = A + \left( \frac{B}{T} \right) \]
where:
- \( A \) and \( B \) are constants.
- \( T \) is the temperature in Kelvin (\( °K \)).
- \( C_{p0} \) has units of \( \text{kJ} / °K \cdot \text{kg} \).
The gas constant for this system is given by:
\[ R = 0.25 \text{kJ} / °K \cdot \text{kg} \]
The constants are:
\[ A = 1.25 \]
\[ B = 173.8029748 \]
We are given two specific temperatures:
- \( T_1 = 300°K \)
- \( T_2 = 400°K \)
#### Objective
Determine the change in specific internal energy (\( u_2 - u_1 \)) in \( \text{kJ} / \text{kg} \). It is important to note that the assumption of constant specific heats is not valid for this calculation.
#### Process
1. **Integrate the specific heat capacity function to determine the change in internal energy:**
Specific heat capacity at constant volume \( C_v \) can be related to \( C_{p0} \) as follows, considering the ideal gas relationship:
\[ C_v = C_{p0} - R \]
2. **Integrate \( C_v \) over the temperature range:**
\[ u_2 - u_1 = \int_{T_1}^{T_2} C_v \, dT \]
\[ \Rightarrow u_2 - u_1 = \int_{T_1}^{T_2} \left( A + \frac{B}{T} - R \right) \, dT \]
Substitute the values of \(A\), \(B\), and \(R\):
\[ u_2 - u_1 = \int_{T_1}^{
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