In this problem, we are asked to calculate the work done on the gas during the compression stroke of a four-stroke engine cycle. The work done in a gas compression or expansion is related to the change in volume and pressure.

We are given:

• Initial volume $V_1 = 445 \, \text{mL}$
• Final volume $V_2 = 75 \, \text{mL}$
• External pressure $P = 13.0 \, \text{atm}$

Formula for work done in compression:

The work done on a gas at constant pressure is calculated by:

$W = -P \Delta V$

where:

• $W$ is the work done on the gas,
• $P$ is the external pressure,
• $\Delta V = V_2 - V_1$ is the change in volume (final volume minus initial volume).

Here the work is negative because the gas is being compressed (work is done on the gas).

Step 1: Convert volumes to liters

We need to convert milliliters (mL) to liters (L), since 1 atm L = 101.3 J.

$V_1 = 445 \, \text{mL} = 0.445 \, \text{L}$ $V_2 = 75 \, \text{mL} = 0.075 \, \text{L}$

Step 2: Calculate the change in volume ($\Delta V$)

$\Delta V = V_2 - V_1 = 0.075 \, \text{L} - 0.445 \, \text{L} = -0.370 \, \text{L}$

Step 3: Apply the formula for work

Now, using the formula $W = -P \Delta V$:

$W = - (13.0 \, \text{atm}) \times (-0.370 \, \text{L})$

$W = 4.81 \, \text{L atm}$

Step 4: Convert the work to joules

To convert $\text{L atm}$ to joules, we use the conversion factor $1 \, \text{L atm} = 101.3 \, \text{J}$:

$W = 4.81 \, \text{L atm} \times 101.3 \, \text{J/L atm}$

$W = 487.253 \, \text{J}$

Step 5: Report the final answer to three significant figures

$W = 487 \, \text{J}$

Final Answer:

The work done on the gas is 487 J.

Would you like a detailed breakdown of any part of this process?

Here are five related questions you might explore:

1. How does the work change if the pressure is doubled?
2. What would the work be if the volumes were in reverse?
3. How does the type of gas affect the work done in such cycles?
4. Can you calculate work done in an adiabatic process?
5. How does this problem relate to the first law of thermodynamics?

Tip: Always check your units when working with gases to ensure consistency, especially when converting between mL, L, atm, and joules.