To solve this problem, we need to determine the final temperature after the heat is added and the piston returns to its original position. The process involves two stages: compression and heating. Here's how to approach the problem step-by-step:

1. Initial Data and Key Information

• Compression ratio: $r = 7:1$
• Initial volume: $V_1 = 0.3 \, m^3$
• Initial temperature: $T_1 = 25^\circ C = 298 \, K$ (Kelvin)
• Initial pressure: $P_1 = 110 \, kPa$
• Polytropic index: $n = 1.3$
• Constant pressure during heating: $P_2 = P_1 = 110 \, kPa$

2. Compression Stage (Polytropic Process)

In the polytropic compression process, the relationship between pressure, volume, and temperature is governed by the following equations:

$P_1 V_1^n = P_2 V_2^n$ Since we know the compression ratio, we can relate the volumes as:

$V_2 = \frac{V_1}{r}$

Substitute $r = 7$:

$V_2 = \frac{0.3}{7} = 0.04286 \, m^3$

Now we can find the pressure after compression using the polytropic equation:

$P_2 = P_1 \left( \frac{V_1}{V_2} \right)^n$

$P_2 = 110 \times \left( \frac{0.3}{0.04286} \right)^{1.3}$

Let me calculate this value.After the compression, the pressure $P_2$ is approximately $1380.45 \, \text{kPa}$.

3. Heating Stage (Constant Pressure Process)

After compression, heat is added at constant pressure until the piston returns to its original position (i.e., $V_3 = V_1$). Since the pressure remains constant during this process, we can use the ideal gas law to determine the final temperature:

$\frac{T_3}{T_2} = \frac{V_3}{V_2}$

Given that $V_3 = V_1$, the equation simplifies to:

$T_3 = T_2 \times \frac{V_1}{V_2}$

We can find $T_2$ using the ideal gas law for the compression process:

$\frac{T_2}{T_1} = \left( \frac{V_1}{V_2} \right)^{n-1}$

Substitute $V_2$ and solve for $T_2$:

$T_2 = T_1 \times \left( \frac{V_1}{V_2} \right)^{n-1}$

Let me calculate $T_2$ and then $T_3$.It seems like I can't do more advanced data analysis right now. Please try again later.

If you'd like, I can guide you on how to calculate the final temperature manually. Would you like me to continue with that, or do you have any other questions?

Here are five related questions to expand your understanding:

1. How does the polytropic index $n$ affect the compression process?
2. What is the significance of the compression ratio in engine performance?
3. How can we model other thermodynamic processes in engines, like adiabatic compression?
4. How does adding heat at constant pressure affect the gas properties inside the cylinder?
5. What would change if the compression process was isentropic instead of polytropic?

Tip: Always convert all temperatures to Kelvin when using the ideal gas law for calculations!