Engineers designing new piston engines and turbines need to understand how work and heat are involved in

Question:

Engineers designing new piston engines and turbines need to understand how work and heat are involved in various compression and expansion cycles. Suppose that 1.00 mol of ideal gas molecules at an initial pressure of 3.00 atm and 292 K expands against a constant external pressure of 0.20 atm from 8.00 L to 20.00 L by two different paths.

(a) Path A is an isothermal, reversible expansion.

(b) Path B, a hypothetical alternative to path A, has two steps. In step 1, the gas is cooled at constant volume until its pressure has fallen to 1.20 atm. In step 2, it is heated and allowed to expand against a constant pressure of 1.20 atm until its volume is 20.00 L and T = 292 K. Determine for each path the work done (w), the heat transferred (q), and the change in internal energy (ΔU).

ANTICIPATE You should expect the value of w to be less negative (less energy lost as work) in the irreversible path and therefore q to be less positive as less energy transferred as heat is needed to maintain the temperature.

PLAN It is a good idea to begin by sketching a plot of each process (FIG. 4B.3).

(a) For an isothermal, reversible expansion, use Eq. 4 of Topic 4A (w = 2nRT ln(V₂/V₁)) to calculate w.

(b) In step 1, the volume does not change, and so no work is done (w = 0). Step 2 is a constant-pressure process, so use Eq. 3 of Topic 4A (w = 2P_e_xΔV) to calculate w. Because internal energy is a state function and because the initial and final states are the same in both paths, ΔU for path B is the same as for path A. Because ΔU = 0 for an isothermal expansion of an ideal gas, in each case, find q for the overall path from ΔU= q 1 w with ΔU = 0. Use 1 L · atm = 101.325 J (Topic 4A) to convert liter-atmospheres into joules.