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?

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?

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 $\gamma$.