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The pV diagram in the figure (Figure 1) shows a cycle of a heat engine that uses 0.250 mole of an ideal gas having γ=1.40. The curved part ab of the cycle is adiabatic. Find the pressure of the gas at point  a
.

To find the pressure of the gas at point \( a \) in this cycle, let's analyze the given information and apply the relevant thermodynamic principles:

1. **Given Data**:
   - **Type of Gas**: Ideal gas
   - **Gas Constant**: \( \gamma = 1.40 \) (adiabatic index, often noted as \( \gamma = \frac{C_p}{C_v} \))
   - **Moles of Gas**: \( n = 0.250 \) moles

2. **Information from Diagram**:
   - The pressure-volume (\( pV \)) diagram shows three points: \( a \), \( b \), and \( c \).
   - Point \( c \) is at \( p = 1.5 \ \text{atm} \) and \( V = 0.0020 \ \text{m}^3 \).
   - Point \( b \) has a volume of \( V = 0.0090 \ \text{m}^3 \).
   - The path from \( a \) to \( b \) is adiabatic, meaning there is no heat transfer during this segment of the cycle.

3. **Using the Adiabatic Process**:
   Since \( a \rightarrow b \) is adiabatic, we can use the adiabatic relationship for an ideal gas:
   \[
   p_a V_a^{\gamma} = p_b V_b^{\gamma}
   \]
   where \( \gamma = 1.40 \).

4. **Finding \( p_a \)**:
   If we know the pressure at \( b \) (which we do not yet have), we can rearrange the equation to solve for \( p_a \):
   \[
   p_a = p_b \left(\frac{V_b}{V_a}\right)^{\gamma}
   \]

To proceed, we need either the pressure at \( b \) or more information about the thermodynamic conditions in the cycle. If there’s additional information, such as the temperature at any point or a method to find \( p_b \), I could calculate \( p_a \) directly.

Would you like further assistance on finding the pressure at \( b \) or any additional details to continue? 

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### Relative Questions:
1. What formula is used to calculate work in an adiabatic process?
2. How would knowing the temperature at any point help in solving for \( p_a \)?
3. What is the significance of the value of \( \gamma \) in an ideal gas process?
4. How is the adiabatic relationship between pressure and volume derived?
5. How does the heat engine's cycle affect the internal energy of the gas?

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### Tip:
For adiabatic processes, knowing either the initial or final pressure and volume can allow you to solve for unknowns, given the specific heat ratio \( \gamma \).

The pV diagram in the figure (Figure 1) shows a cycle of a heat engine that uses 0.250 mole of an ideal gas having γ=1.40. The curved part ab of the cycle is adiabatic. Find the pressure of the gas at point a.

Calculate the pressure of an ideal gas at point a in an adiabatic heat engine cycle with 0.250 mole of gas and γ=1.40. This problem uses thermodynamic principles including the adiabatic process and ideal gas law.

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