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Chapter 18: Problem 42

A gas expands isothermally from state \(A\) to state \(B,\) in the process absorbing 35 J of heat. It's then compressed isobarically to state \(C,\) where its volume equals that of state \(A .\) During thecompression, \(22 \mathrm{J}\) of work are done on the gas. The gas is then heated at constant volume until it returns to state \(A\). (a) Draw a \(p V\) diagram for this process. (b) How much work is done on or by the gas during the complete cycle? (c) How much heat is transferred to or from the gas as it goes from \(B\) to \(C\) to \(A\) ?

Short Answer

Expert verified
The total work done on or by the gas during the complete cycle is 13J and the total heat transferred from B to C to A is 13J.

Step by step solution

01

Work Calculation from A to B

For an isothermal process, work \(W_{AB}\) can be calculated by the formula \(W = Q = 35 \mathrm{J}\), where Q stands for the absorbed heat.
02

Work calculation from B to C

In an isobaric process, the work \(W_{BC}\) done is given by the formula \(W = P \Delta V = 22 \mathrm{J}\), where P is the constant pressure and \(\Delta V\) is the change in volume. Since it is work done on the gas, it is negative: \(W_{BC} = - 22 \mathrm{J}\)
03

Work calculation from C to A

In an isochoric process, the volume is constant. Therefore, the work done \(W_{CA}\) is zero because \(W = P \Delta V\);since \(\Delta V = 0\), \(W_{CA} = 0 \mathrm{J}\).
04

Total work calculation

The total work done in the cycle is the sum of the work done in each process: \(W_{total} = W_{AB} + W_{BC} + W_{CA} = 35 \mathrm{J} - 22 \mathrm{J} + 0 \mathrm{J} = 13 \mathrm{J}\).
05

Heat transfer calculation

The total heat transferred over the entire process is the heat transferred in process AB and the heat transferred in process BC. The heat transferred in CA is zero because there is no volume change so no work is done. The heat transferred in BC (isobaric process) is \(Q_{BC} = \Delta U + W_{BC}\) where \( \Delta U = 0 \) because it is a cyclic process and so internal energy change is zero. Thus, \(Q_{BC} = 0 \mathrm{J} - 22 \mathrm{J} = -22 \mathrm{J}\), where '-' sign indicates that heat was transferred from the gas. Therefore, total heat transferred is \(Q_{total} = Q_{AB} + Q_{BC} = 35 \mathrm{J} - 22 \mathrm{J} = 13 \mathrm{J}\).

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Most popular questions from this chapter

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