c) Use dU = 8Q-PdV to compute the heat, QD¬A, associated to the process D→ A along the black vertical line shown in the figure above. d) Calculate the line integral of dU along the path shown below. P- 3 P1 2 P1 P1 V V1 2V1 Problem 3d
c) Use dU = 8Q-PdV to compute the heat, QD¬A, associated to the process D→ A along the black vertical line shown in the figure above. d) Calculate the line integral of dU along the path shown below. P- 3 P1 2 P1 P1 V V1 2V1 Problem 3d
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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I need help with the homework. I want detailed steps to each part of the problem. Solve for C and D part
![**Problem 3a-3c**
For a mole of a perfect monoatomic gas, the internal energy, \( U \), can be expressed as a function of the pressure and volume as
\[ U = U(P, V) = \frac{3}{2} PV \]
a) Calculate *explicitly* the line integral of \( dU \) along the closed path \( ABCD \) shown as a black trace in the \( P-V \) graph below.
**Graph Explanation:**
- The graph is a \( P-V \) (Pressure-Volume) diagram.
- The axes are labeled with pressure \( P \) on the vertical axis and volume \( V \) on the horizontal axis.
- The path \( ABCD \) forms a closed loop made up of straight lines and a curved section:
- **Segment AB:** Horizontal from \( (V_1, P_1) \) to \( (6V_1, P_1) \) at constant pressure \( P_1 \).
- **Segment BC:** Diagonal from \( (6V_1, P_1) \) to \( (4V_1, 3P_1) \), depicting a change in both pressure and volume.
- **Segment CD:** Horizontal from \( (4V_1, 3P_1) \) to \( (V_1, 3P_1) \) at constant pressure \( 3P_1 \).
- **Segment DA:** Vertical from \( (V_1, 3P_1) \) back to \( (V_1, P_1) \) at constant volume \( V_1 \).
- The red curve between points \( B \) and \( C \) is marked with segments \( h \) and \( s \).
This closed path in the \( P-V \) diagram is used to evaluate the integral for change in internal energy around the cycle.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f6272ad-1a25-4a67-b389-4d6bf92e8dd2%2Fa91ab7e6-b7e9-417a-9313-e4d5f39e3f63%2Flj3jcia_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 3a-3c**
For a mole of a perfect monoatomic gas, the internal energy, \( U \), can be expressed as a function of the pressure and volume as
\[ U = U(P, V) = \frac{3}{2} PV \]
a) Calculate *explicitly* the line integral of \( dU \) along the closed path \( ABCD \) shown as a black trace in the \( P-V \) graph below.
**Graph Explanation:**
- The graph is a \( P-V \) (Pressure-Volume) diagram.
- The axes are labeled with pressure \( P \) on the vertical axis and volume \( V \) on the horizontal axis.
- The path \( ABCD \) forms a closed loop made up of straight lines and a curved section:
- **Segment AB:** Horizontal from \( (V_1, P_1) \) to \( (6V_1, P_1) \) at constant pressure \( P_1 \).
- **Segment BC:** Diagonal from \( (6V_1, P_1) \) to \( (4V_1, 3P_1) \), depicting a change in both pressure and volume.
- **Segment CD:** Horizontal from \( (4V_1, 3P_1) \) to \( (V_1, 3P_1) \) at constant pressure \( 3P_1 \).
- **Segment DA:** Vertical from \( (V_1, 3P_1) \) back to \( (V_1, P_1) \) at constant volume \( V_1 \).
- The red curve between points \( B \) and \( C \) is marked with segments \( h \) and \( s \).
This closed path in the \( P-V \) diagram is used to evaluate the integral for change in internal energy around the cycle.
Transcribed Image Text:**Problem Statement:**
c) Use the equation \( dU = \delta Q - PdV \) to compute the heat \( Q_{D \to A} \) associated with the process from \( D \) to \( A \), along the black vertical line shown in the figure above.
d) Calculate the line integral of \( dU \) along the path shown below.
**Graph Explanation:**
- The graph is a Pressure-Volume (P-V) diagram.
- The y-axis represents pressure \( P \) with marked levels: \( P1 \), \( 2P1 \), and \( 3P1 \).
- The x-axis represents volume \( V \) with marked levels: \( 0 \), \( V1 \), and \( 2V1 \).
- The path shown is a spiral red line starting from a point at \( (V1, P1) \) and spiraling outwards towards a higher volume and pressure.
- The final point of the spiral appears to be between \( 2V1 \) and the lower pressure levels.
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