Explanation Write the given definition for temperature is as follows. T = ( ∂ u ∂ s ) ν (I) Here, the temperature is T , the internal energy is u , the entropy is s , the volume is ν , and the symbol ∂ indicates the partial differential of variables. Consider the given system is thermally isolated. Write the formula for change in net entropy of the system. d S system = m A d s A + m B d s B (II) Here, the change in entropy of the system A is d s A and the change in entropy of the system B is d s B . Consider the entropy ( S ) is the function of internal energy ( u ) and the volume ( ν ) . s = s ( u , v ) The total derivative of the entropy is expressed in terms of partial derivatives of u , ν as follows. d s = ( ∂ s ∂ u ) ν d u + ( ∂ s ∂ ν ) u d v (III) It is given that the volume of the system is constant. Hence, the change in volume is zero. ν = constant d ν = 0 Substitute 0 for d ν in Equation (III). d s = ( ∂ s ∂ u ) ν d u + ( ∂ s ∂ ν ) u ( 0 ) = ( ∂ s ∂ u ) ν d u (IV) Rewrite the equation (I) as follows by dividing d u by Equation (I) on both sides. d u T = d u ( ∂ u ∂ s ) ν d u T = ( ∂ s ∂ u ) ν d u (V) Substitute d s for ( ∂ s ∂ u ) ν d u in Equation (V). d u T = d s (VI) Express the Equation (VI) for system A and system B as follows. d s A = d u A T A d s B = d u B T B Substitute d u A T A for d s A and d u B T B for d s B in Equation (II)

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Entropy

In physics, entropy means the amount of disorder or the degree of randomness associated with a thermodynamic system. The system is not self-organized when the entropy of the system is high. For instance, when a system is at an elevated temperature, its m…

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Chapter 12.6, Problem 49P

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To prove that the net entropy change of two constant-volume systems filled with simple compressible substances is zero as the two systems approach thermal equilibrium.

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Chapter 12.6, Problem 48P

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To prove that the net entropy change of two constant-volume systems filled with simple compressible substances is zero as the two systems approach thermal equilibrium.

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To prove that the net entropy change of two constant-volume systems filled with simple compressible substances is zero as the two systems approach thermal equilibrium.

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Chapter 12 Solutions