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Let's break down the given problem step by step to solve it:

### Information provided:
- Volume at state A, \( V_A = 250 \, \text{cm}^3 = 2.5 \times 10^{-4} \, \text{m}^3 \)
- Pressure at state A, \( P_A = 100 \, \text{kPa} \)
- Temperature at state A, \( T_A = 300 \, \text{K} \)
- Transformation sequence:
  1. Isobaric expansion to \( V_B = 500 \, \text{cm}^3 = 5.0 \times 10^{-4} \, \text{m}^3 \) (State B)
  2. Isochoric increase in pressure to \( P_C = 200 \, \text{kPa} \) (State C)
  3. Isobaric contraction back to \( V_D = 250 \, \text{cm}^3 \) (State D)
  4. Isochoric decrease in pressure back to state A (State A)

We'll solve each part in order:

### (a) Sketch the P–V diagram

1. **State A to B**: Isobaric expansion from \( V_A \) to \( V_B \). The pressure remains constant at 100 kPa, and the volume increases.
2. **State B to C**: Isochoric (constant volume) transformation, so volume remains at 500 cm³, and pressure increases to 200 kPa.
3. **State C to D**: Isobaric contraction from \( V_B \) to \( V_D \). Pressure remains constant at 200 kPa, and the volume decreases.
4. **State D to A**: Isochoric decrease in pressure at constant volume from 200 kPa to 100 kPa.

### (b) Use the ideal gas equation to calculate the temperature at states B, C, and D

We will use the ideal gas law: 
\[
PV = nRT
\]
Where \( P \) is pressure, \( V \) is volume, \( T \) is temperature, and \( R \) is the gas constant (\( 8.314 \, \text{J/mol K} \)).

1. **At state B** (isobaric expansion):
   \[
   P_A V_A / T_A = P_B V_B / T_B \quad \text{(because pressure is constant, use \( V_B \) and solve for \( T_B \))}
   \]

2. **At state C** (isochoric transformation):
   \[
   \frac{P_B}{T_B} = \frac{P_C}{T_C} \quad \text{(since volume is constant, solve for \( T_C \))}
   \]

3. **At state D** (isobaric contraction):
   \[
   P_C V_C / T_C = P_D V_D / T_D \quad \text{(similar to previous steps, solve for \( T_D \))}
   \]

### (c) Calculate the work done by the gas

For **isobaric** processes, the work done by the gas is given by:
\[
W = P \Delta V
\]
We'll calculate the work done during the isobaric transformations A to B and C to D.

### (d) Calculate the work done on the gas

Since work is done **on** the gas during the contraction (State C to D), this will involve a similar calculation using the isobaric work formula.

### (e) Net work done during one cycle

The net work done is the sum of the work done by the gas and the work done on the gas. For a complete cycle, this can also be represented as the area enclosed by the P–V curve in the diagram.

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I will proceed with the calculations for parts (b), (c), (d), and (e).It seems like I can’t do more advanced data analysis right now. Please try again later. However, we can walk through the manual steps to perform these calculations if you'd like.

Would you like to proceed with manual explanations, or have any specific questions about one of the steps we discussed? 

Here are five related questions you can explore further:
1. How do isochoric and isobaric processes differ in thermodynamics?
2. What is the significance of the area under the curve in a P–V diagram?
3. Can you derive the ideal gas law from first principles?
4. What is the difference between work done by the gas and on the gas?
5. How does the number of moles of gas affect the ideal gas law calculations?

**Tip**: In thermodynamic cycles, the net work done by the gas over one cycle is always equal to the area enclosed by the P-V diagram.

250 cm³ of gas at 300 K exerts a pressure of 100 kPa on its container (state A). It undergoes the following cycle of transformations: 1. an isobaric expansion to 500 cm³ (state B), 2. an isovolumetric transformation to a pressure of 200 kPa (state C), 3. an isobaric contraction back to 250 cm³ (state D), 4. an isovolumetric transformation back to state A. (a) Sketch a P–V diagram representing this cycle, labeling the states A, B, C, and D. (b) Use the ideal gas equation to calculate the temperature at B, C, and D. (c) Calculate the amount of work done by the gas. (d) Calculate the amount of work done on the gas. (e) What is the net work done during one cycle?

Explore the thermodynamic cycle of an ideal gas undergoing isobaric and isovolumetric transformations. This problem includes the calculation of work done, temperature changes using the ideal gas law, and analysis of the P–V diagram. Suitable for college-level students.

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