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Work Done in a Quasi-Static Path

Let us consider a piston-cylinder with gas mechanism as shown in Fig. 3.4 (a). The gas is in thermodynamic equilibrium at position (1) which is described by the state (P1, V1). Under the pressure P1 the piston moves slowly to position (2) which is also equilibrium

state described by (P2, V2). Any intermediate stage is also in equilibrium. And at any

intermediate point (P, V), the piston moves further by infinitesimal distsmce dl. If A is th

area of cross section of the cylinder, the volume displaced is AdZ = dV. The small work

done is F.dl.

dW  = f.dl = PAdl = PdV

 

The volume changes from V1 to V2, therefore the total work done is given by

The magnitude of work can be calculated also from the area under the process (quasistatic) 1..:2 as shown in Fig. 3.4 (b) by shaded area.

 

3.3.1 Path Function and Point Function

 

Let A, B, and C all are quasi-static paths to reach to state 2 from state 1. As per equation (3.1), the work is the area under  each quasi-static process. Since the paths are different the area under each path will be different. So it is evident that the amount of workdone during each process not only is a function of the end states of the process, but also depends on the path that is followed in going from one state to another. For this reason work is called a path function or in methematical term aw is an inexact differential. As thermodynamic property is having a fixed point in each state in the diagram (Fig. 3.5), it is called point function. The differentials of point functions are exact differentials, and the integration is simply

Thus we can say V1 is the volume at 1, V2 is the volume at 2, and the change in

volume depends only on the initial and final $tates. We have used a, (a cut on the tip of d) to denote inexact differentials.

 

PdV Work in Various Quasi-Static Processes

 

(a) Constant volume process : Here V = Const.

Thus ,                          dV  = 0 , so

(b) Constant pressure process :Here P = Const. so we can take out P from integration.

(c) Process in which PV =canst. = c.

or ,                          w1 -2 = p1V1 In V2 / V1 = P1V1 In P1 / P2

 

(d) Process in which PV” = Const. = C

C = (V –n+1/ -n +1) v1 v2 = C / 1-n [V12  - V1-n]

= 1 / 1-n [ CV1-n 2 – CV 1-n 1 ]

= 1/ 1-n [ P2 V2 V1 – P1 V1 V1-n]

W1-2 = 1 / 1-n [ P2 V2 – P1 V1]

= P1V1 – P2 V2 / n -1 = P1 V1 /n-1 [ 1-(P2 P1) n-1 / n

If the system is having ideal gas, then we have

P1 V1 / T1 = P2 V2 / T2

We get                              T2 / T1 = P2 V2 / P1 V1 = ( P2 / P1 ) n-1 /n = ( V1 / V2) n-1

W1-2  = mR (T2 –T1 / 1-n) , [ P2 V2 = mRT2 , P1 V1 = mRT1 ]