Let's re-calculate $P_2$ to ensure accuracy. The process from state 1 to state 2 is adiabatic. For an adiabatic process, the relationship between pressure and volume is given by: $P_1 \times V_1^\gamma = P_2 \times V_2^\gamma$ We can rearrange this formula to solve for $P_2$: $P_2 = P_1 \times (V_1 / V_2)^\gamma$ Now, let's substitute the known values: $P_1 = 100 \text{ kPa}$ $V_1 = 0.082 \text{ m}^3$ $V_2 = 0.0054 \text{ m}^3$ $\gamma = 1.4$ Step 1: Calculate the ratio of volumes. $V_1 / V_2 = 0.082 \text{ m}^3 / 0.0054 \text{ m}^3$ $V_1 / V_2 \approx 15.185$ Step 2: Raise the volume ratio to the power of $\gamma$. $(V_1 / V_2)^\gamma = (15.185)^{1.4}$ Let's calculate this value: $(15.185)^{1.4} \approx 45.0804$ Step 3: Multiply by $P_1$. $P_2 = 100 \text{ kPa} \times 45.0804$ $P_2 \approx 4508.04 \text{ kPa}$ So, your calculation of $P_2 = 4508.043827 \text{ kPa}$ is correct. My previous value might have had a slight rounding difference. The pressure at state 2 is approximately 4508.04 kPa.