السلام عليكم ورحمة الله وبركاته
The total entropy S of two independent systems given by S = S1 +S2 ,
on other hand the thermodynamics of the system is M =M1*M2
Assume S is unknown function of M ; S= J(M) in such way:
J(M1) + J(M2) = J (M1*M2) .................................[1]
I. Take partial derivative of equation [1] with respect to M1
(
dJ(M1))/dM1 = (dJ(M1*M2))/(d(M1*M2))*(d(M1*M2))/dM1
(dJ(M1))/dM1 = M2 * (dJ(M1*M2))/(d(M1*M2)) ……………………………….[2]
II. Take partial derivative of equation [1] with respect to M2
(dJ(M2))/dM2 = (dJ(M1*M2))/(d(M1*M2))*(d(M1*M2))/dM2
(dJ(M2))/dM2 = M1 * (dJ(M1*M2))/(d(M1*M2)) ……………………………….[3]
From [2] &[3]
M1 (dJ(M1))/dM1 = M2 (dJ(M2))/dM2 ………………………………….[4]
Each term in 4 equal constant (KB) KB=M (dJ(M))/dM
[align=center]dJ(M)= KB dM/M ; Integrate both side J(M) = KB log (M)
S = KB log (M) [/align]
:laughter01: