Variations in Entropy
Changes in Entropy that occur when an Ideal Gas is subjected to both reversible and irreversible operations.
The Variations of Entropy For a Perfect Gas,
$$\delta \Phi = \frac{dq}{T}\;for\;a\;reversible\;operation$$
(1)
$$\delta \Phi \;T = \delta\,q$$
(2)
At Constant Volume:-
$$\delta \Phi = \frac{w\,C_V\,dt}{T}$$
(3)
$$\therefore\;\;\;\;\;\Phi _2 - \Phi _1 = \int_{T_1}^{T_2}\frac{w\;C_V\,dt}{T}$$
(4)
$$= wC_V\;Ln\frac{T_2}{T_1}\;\;\;\;\;or\;\;\;\;\;w\;C_V\;ln\frac{P_2}{P_1}$$
(5)
At Constant Pressure:-
$$\delta \Phi = \frac{w\;C_P\;dt}{T}$$
(6)
$$\therefore\;\;\;\;\;\Phi _2 - \Phi _1 = \int_{T_1}^{T_2}\;\frac{w\;C_P\;dt}{T}$$
(7)
$$= w\;C_P\;Ln\frac{T_2}{T_1}\;\;\;\;\;or\;\;\;\;\;w\;C_P\;Ln\frac{V_2}{V_1}$$
(8)
For an Isothermal Operation:-
$$Heat\;Supplied = Work\;Done + \delta\,U\,( Constant)$$
(9)
$$= P\;\delta \,V$$
(10)
$$\therefore\;\;\;\;\;\delta \Phi = \frac{p\delta V}{T} = w\;R\;\frac{dv}{V}$$
(11)
$$Thus\;\;\;\;\;\Phi _2 - \Phi_1 = w\,R\,Ln\frac{V_2}{V_1}\;\;\;\;\;or\;\;\;\;\;w\,R\,Ln\frac{P_1}{P_2}$$
(12)
Entropy for an irreversible Operation
Consider a gas changing from $P_1V_1T_1$ to $P_2V_2T_2$. The Function of State depends upon the values of P,V,&T at 1 & 2, and it is independent of the process.
Stage 1
$$\Phi _a - \Phi _1 = wC_V\;Ln\;\frac{T_a}{T_1}\;\;\;\;\;or\;\;\;\;\;wC_V\;Ln\;\frac{P_2}{P_1}$$
(13)
$$\Phi _2 - \Phi _a = wC_P\;Ln\;\frac{T_2}{T_a}\;\;\;\;\;or\;\;\;\;\;wC_P\;Ln\;\frac{V_2}{V_1}$$
(14)
$$\therefore\;\;\;\;\;\Phi _2 - \Phi _1 = w\left( C_V\;Ln\;\frac{P_2}{P_1} + C_P\;Ln\;\frac{V_2}{V_1} \right)$$
(15)
Sketching Operations on P.V. and T. Phi graphs
NOTE
The Area under the curve on a P.V. graph is the Work Done.
The Area under the curve on a T.$\Phi$ graph is the heat Supplied.
Constant Volume 1 - 2
$$P\;V^n = Constant$$
(16)
$$V = Constant$$
(17)
$$n = infinity$$
(18)
$$\Phi _2 - \Phi _1 = w\;C_V\;Ln\;\frac{T_2}{T_1}$$
(19)
$$\therefore\;\;\;\;\;\Phi = w\;C_V\;Ln\;T$$
(20)
Constant Pressure 1 - 3
$$P\;V^n = Constant$$
(21)
$$P = Constant$$
(22)
$$n = 0$$
(23)
$$\Phi _2 - \Phi _1 = wC_P\;Ln\;\frac{T_2}{T_1}$$
(24)
Isothermal 1 - 4
$$P\;V^n = Constant$$
(25)
$$T = Constant$$
(26)
$$\therefore\;\;\;P\;V = Constant$$
(27)
$$n = 1$$
(28)
Adiabatic 1 - 5
$$n = \gamma$$
(29)
$$\Phi = Constant$$
(30)