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To solve part (a) and find \( P_x \) in atm for the given thermodynamic cycle, let’s analyze the processes in detail.

### Given dataTo solve part (a) and find \( P_x \) in atm for the given thermodynamic cycle, let’s analyze the processes in detail.

### Given 
- **Initial pressure** \( P_0 = 11.39 \, \text{atm} \)
- **Initial volume** \( V_1 = 10.20 \, \text{L} \)
- **Final volume** \( V_2 = 41.10 \, \text{L} \)
- For a **monatomic ideal gas**:
  - \( C_V = \frac{3}{2} R \)
  - \( C_P = \frac{5}{2} R \)
  - \( R = 8.31 \, \frac{\text{J}}{\text{mol} \cdot \text{K}} \)
  - **Conversion**: \( 1 \, \text{atm} \cdot \text{L} = 101 \, \text{J} \)

### Step-by-Step Solution:
The question seems to involve two processes in the cycle:
1. **Adiabatic Process** (no heat exchange, \( Q = 0 \))
2. **Isothermal Process** (constant temperature)

The required pressure \( P_x \) corresponds to the **adiabatic expansion** from \( (P_0, V_1) \) to \( (P_x, V_2) \).

### Adiabatic Process for an Ideal Gas:
In an adiabatic process, the relation between pressure and volume is given by:
\[
P_0 V_1^\gamma = P_x V_2^\gamma
\]
where \( \gamma = \frac{C_P}{C_V} = \frac{5/2 \, R}{3/2 \, R} = \frac{5}{3} \).

Rearranging to solve for \( P_x \):
\[
P_x = P_0 \left(\frac{V_1}{V_2}\right)^\gamma
\]

#### Substituting Values:
\[
P_x = 11.39 \, \text{atm} \times \left(\frac{10.20 \, \text{L}}{41.10 \, \text{L}}\right)^{5/3}
\]

Let's calculate this value.The calculated pressure \( P_x \) is approximately \( 1.12 \, \text{atm} \).

Would you like a detailed explanation of each step? Also, here are some related questions that could extend this topic:

1. How would the process change if the gas were diatomic instead of monatomic?
2. What is the work done during the adiabatic process?
3. How can we calculate the temperature change during the adiabatic expansion?
4. What would happen to \( P_x \) if the volume \( V_2 \) were smaller?
5. How is the efficiency of this thermodynamic cycle determined?

**Tip:** Remember, for adiabatic processes in ideal gases, the relationship between pressure and volume depends on the specific heat capacities through the adiabatic index \( \gamma \).

A heat engine operates upon the following thermodynamic cycle. Given P0 = 11.39 atm, V1 = 10.20 L, and V2 = 41.10 L, determine the pressure Px in atm during the adiabatic expansion process. The gas is monatomic, with Cv = (3/2)R and Cp = (5/2)R.

Calculate the pressure Px during the adiabatic expansion in a thermodynamic cycle, given initial conditions P0 = 11.39 atm, V1 = 10.20 L, and V2 = 41.10 L. Understand the concepts of ideal gas laws and adiabatic processes for monatomic gases.

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