The PV diagram in the figure below shows a set of thermodynamic processes that m

Question

The PV diagram in the figure below shows a set of thermodynamic processes that make up a cycle ABCDA for a monatomic gas, where AB is an isothermal expansion occurring at a temperature of 390 K. There are 1.80 mol of gas undergoing the cycle with

PA = 1.01 106 Pa,

PB = 5.15 105 Pa,

and

PC = 2.02 105 Pa.

(a) Find the volumes VA and VB.

(b) Find the work done in each process of the cycle and the total work done for the whole cycle.

(c) Find the change in thermal energy during the constant-volume process BC.
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Explanation / Answer

(a)

We will use the ideal gas equation;

PV = nRT,

AB is an isothermal process so temeprature from A to B will be same at T = 390K

Using ideal gas equation at A we get,

PAVA = nRT

or VA = nRT/PA = (1.8 mol)(8.31 J/mol-K)(390 K)/(1.01X106 Pa)

or VA = 5.775X10-3 m3

Similarly,

VB = nRT/PB = (1.8 mol)(8.31 J/mol-K)(390 K)/(0.515X106 Pa)

or VB = 11.327X10-3 m3

***************************************************************************************************

(b)

AB is an isothermal process and work done in an isothermal process is given by;

WAB = nRTln(VB/VA) = (1.8 mol)(8.31 J/mol-K)(390 K)ln[(11.327X10-3 m3)/(5.775X10-3 m3)]

or WAB = 3929.82 J

In process BC, volume change is zero so work done is zero.

So, WBC = 0 J

Process CD is an isobaric process and work done is given by,

WCD = PC(VD - VC) = (2.02X105 Pa)(VA - VB) = (2.02X105 Pa)(5.775X10-3 m3 - 11.327X10-3 m3)

or WCD = -1121.5 J

In process DA, volume change is zero so work done is zero.

So, WDA = 0 J

Wtot = WAB + WBC + WCD + WDA = 3929.82 J + 0 - 1121.5 J + 0

or, Wtot = 2808.32 J

***************************************************************************************************

(c)

During BC the volume remains constant so we have;

PB/TB = PC/TC

or TC = PCTB/PB = (2.02X105 Pa)(390 K)/(0.515X106 Pa)

or TC = 152.97 K

change in thermal energy during BC is,

UBC = (3/2)nR(TC - TB) = (1.5)(1.8 mol)(8.31 J/mol-K)(152.97 K - 390 K)

or UBC = -5318.22 J

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This concludes the answers. Check the answer and let me know if it's correct. If you need any more clarification or correction, feel free to ask.....

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