Question 1

Isothermal compression, n = 0.5 mol, T = 600 K, so nRT = 300R. Volumes: V₁ = 300R/2 = 150R, V₂ = 300R/P, V₃ = 300R/8 = 37.5R. Work done on gas: |W| = P(V₁ − V₂) + 8(V₂ − V₃) = P·150R − 300R + 2400R/P − 300R = 150R·P + 2400R/P − 600R. Minimize: d/dP → 150R = 2400R/P² → P² = 16 → P = 4 bar (within 2 < P < 8). |W|min = 600R + 600R − 600R = 600R.

**Calculating simple expressions** I need to simplify the calculation involving the formula nRT300R. Specifically, I’m working with W=300R(P/2-1+8/P-1), which simplifies to W=300R(P/2+8/P-2). It looks like I have to find the minimum value of P4, using the expression 300R(2+2-2) to reach 600R. Oh, and I need to make sure I output the result in just JSON format as the user requested!

**Solving the gas work problem** I need to solve this by providing a final JSON output. The question is about an ideal gas with n=0.5 mol, initially at 2 bar and 600K, being compressed in two steps: first against a constant external pressure (P bar) where 2 < P < 8, and then against 8 bar. I'll find the total work done (W) on the gas and analyze the sign convention carefully, as the work done on the gas is usually positive. Let's compute this and ensure to understand the sign convention properly! **Calculating work for ideal gas** I need to ensure I'm using the right units: pressure in bar and work in Joules. The work done is calculated with the equation Work = P_ext ΔV. Since I'm working with an ideal gas at 600K, I can find nRT at 0.5 mol, which comes to 300R J. In the first step, the initial pressure is 2 bar, and I set P_ext equal to P. The work done on the gas is then calculated as 300R(P/2 - 1). For the second step, with an initial pressure of P and a final pressure of 8 bar, I compute the work using P_ext = 8, leading to a total work of 300R(P/2 + 8/P - 2). **Calculating minimum work done on gas** I'm working through the equation P^2=16, which gives P=4 in the interval. When I plug that into the function, f(4) results in 2. The work done, W, equals 600R, confirming option B. I need to check the endpoints for P values approaching 2 and 8. They also yield a minimum work value of 600R. It's crucial to ensure everything is represented clearly in JSON format and to consider the thermodynamics conventions regarding work done on gas.

I'm working through the two-step irreversible compression, calculating the work done in step 1 as gas moves from 2 bar to intermediate pressure P against constant external pressure, then setting up the same approach for step 2 from P to the final pressure. Adding the two work expressions together, I get total work as a function of P alone, and minimize it by taking the derivative and setting it to zero, which gives P=4 bar as the optimal intermediate pressure. Plugging that back in yields total work of 600R, matching option B. **Reasoning (brief):** For the two-step irreversible compression: - Step 1 (2 bar → P bar, ext. pressure = P): W₁ = nRT(P/2 − 1) - Step 2 (P bar → 8 bar, ext. pressure = 8): W₂ = nRT(8/P − 1) Total: W = nRT[P/2 + 8/P − 2] Minimize f(P) = P/2 + 8/P − 2 for 2 < P < 8: f'(P) = 1/2 − 8/P² = 0 → P = 4 (interior minimum) f(4) = 2 + 2 − 2 = 2 W_min = nRT × 2 = 0.5 × R × 600 × 2 = 600R ```json