Final answer:
The work done by carbon dioxide gas in a piston-cylinder assembly is determined by calculating the area under the pressure-volume curve, which is represented by a linear relationship in this case. The work is the area of a trapezoid on a PV diagram and is calculated using initial and final volumes and pressures, then converted from ft-lbf to Btu for the final value.
Step-by-step explanation:
Work Done During Compression
To determine the work done by the carbon dioxide gas during the process in a piston-cylinder assembly, we use the given pressure-volume relationship p = 23.75 - 7.5V. Since work is the area under the pressure-volume (PV) curve for the process on a PV diagram, and we are given a linear relationship, we can calculate the work done as the area under this line between the initial and final volumes. This is equivalent to finding the area of a trapezoid formed by the initial and final states on a PV diagram.
The initial state has a volume (V1) of 2.5 ft3 and an initial pressure (P1) given by substituting V1 into the relationship, resulting in P1 = 23.75 - 7.5(2.5) lbfin2. Similarly, the final state has a volume (V2) of 0.5 ft3 and by substituting V2 into the equation we find P2 = 23.75 - 7.5(0.5) lbfin2.
The work done is the integral of PdV from V1 to V2, which for a linear relationship is (P1 + P2)/2 * (V1 - V2). We then convert the units of work from ft-lbf to Btu to get the final answer.