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To solve this problem, let's focus on each part and gather the given dataTo solve this problem, let's focus on each part and gather the given 

### Given Data:
1. **Initial Pressure** (\( P_0 \)) = 10.74 atm
2. **Volume at Point 1** (\( V_1 \)) = 7.90 L
3. **Volume at Point 2** (\( V_2 \)) = 37.30 L
4. The gas is **diatomic**, which implies the degrees of freedom:
   - \( C_V = \frac{5}{2}R \)
   - \( C_P = \frac{7}{2}R \)
5. Universal gas constant, \( R = 8.31 \, \frac{\text{J}}{\text{mol K}} \)
6. \(1 \text{ atm} \times 1 \text{ L} = 101 \, \text{J}\)

### Objective:
**Find**:
1. **a)** The pressure \( P_x \) during the **adiabatic process** from \( V_1 \) to \( V_2 \).

### Solution Strategy:
Since we have an **adiabatic process** for a diatomic ideal gas, we can use the adiabatic relationship:

\[
P_1 V_1^{\gamma} = P_2 V_2^{\gamma}
\]

where:
- \( \gamma = \frac{C_P}{C_V} = \frac{7}{5} \) for a diatomic gas.

Let’s calculate \( P_x \):

1. **Step 1**: Substitute values into the adiabatic relationship equation:
   \[
   P_x = P_0 \left(\frac{V_1}{V_2}\right)^{\gamma}
   \]

2. **Step 2**: Plug in \( P_0 = 10.74 \, \text{atm} \), \( V_1 = 7.90 \, \text{L} \), \( V_2 = 37.30 \, \text{L} \), and \( \gamma = 1.4 \).

Would you like me to calculate this for you? 

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### Follow-Up Questions:

1. How does an adiabatic process differ from an isothermal process in thermodynamics?
2. What role does the specific heat ratio (\( \gamma \)) play in adiabatic processes?
3. How would the process change if the gas were monatomic instead of diatomic?
4. Can this type of problem be extended to real gases, and what adjustments would be needed?
5. How is the work done in an adiabatic process calculated?

**Tip:** Remember, in adiabatic processes, no heat is exchanged with the surroundings, meaning \( Q = 0 \).

A heat engine operates upon the following thermodynamic cycle. Consider P0 = 10.74 atm, V1 = 7.90 L, and V2 = 37.30 L. Find the pressure Px in atm during the adiabatic process. Given that the gas is diatomic.

Solve for the pressure Px during the adiabatic process of a thermodynamic cycle where P0 = 10.74 atm, V1 = 7.90 L, and V2 = 37.30 L. The problem uses key thermodynamic concepts, including the adiabatic process and ideal gas law, tailored for students in Grades 11-12.

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