By C. Truesdell
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Extra resources for The Concept and Logic of Classical Thermodynamics As a Theory of Heat Engines. Rigorously Constructed Upon Foundations Laid by S. Carnot and F. Reech
The differential of this function is dF = dU − T dS − SdT = − pdV − SdT. 12) The independent variables are now V and T , and we ﬁnd p=− ∂F ∂V , T S=− ∂F ∂T . 13), we obtain U = F−T ∂F ∂T = −T 2 ∂ F T . 15) V This is called the Gibbs–Helmholtz equation. The fourth quantity to be discussed is the Gibbs potential, which is deﬁned by G = U − T S + pV = F + pV = H − T S. 16) Whichever form of the deﬁnition we take, the differential of G is given by dG = −SdT + V d p. 17) It is seen that the independent variables are changed into T and p, and the following relations are obtained: S= ∂G ∂T , V = p ∂G ∂p .
Is the ideal gas absolute temperature. Any real gas behaves very much like an ideal gas as long as the mass density is sufﬁciently small. Let us assume that the Carnot cycle is made up of the following four stages: Stage (i) The ideal gas is initially prepared at state A( p0 , V0 , 1 ). The gas is then isolated from the heat bath and compressed adiabatically until the temperature of the gas reaches 2 . At the end of this process the gas is in state B( p1 , V1 , 2 ). Stage (ii) The gas is brought into thermal contact with a heat bath at temperature 2 and it is now allowed to expand while the temperature of the gas is kept at 2 until the gas reaches the state C( p2 , V2 , 2 ).
Let us suppose that two Carnot cycles are operated in a series combination as is shown in Fig. 4. Then, from the preceding argument, Q2 = f (θ1 , θ2 ), Q1 Q1 = f (θ0 , θ1 ). 66) If we look at the combination of the cycle C, heat bath R1 , and cycle C as another Carnot cycle, we have Q2 = f (θ0 , θ2 ). 67) This means f (θ1 , θ2 ) = f (θ0 , θ2 ) . f (θ0 , θ1 ) θ2 R2 Q2 ✗✔ ❄ C ✖✕ Q1 ❄ θ1 R1 Q1 ✗✔ ❄ C ✖✕ Q0 ❄ R0 θ0 Fig. 4. 68) 18 1 The laws of thermodynamics Since the left hand side of the equation does not depend upon θ0 , the right hand side of the equation is not allowed to contain θ0 .
The Concept and Logic of Classical Thermodynamics As a Theory of Heat Engines. Rigorously Constructed Upon Foundations Laid by S. Carnot and F. Reech by C. Truesdell