# Lecture_4 - Part 2 States of Pure Substances in Terms of...

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Part 2 — States of Pure Substances in Terms of Measurables Lecture 4 — PvT surfaces for pure components Gibbs phase rule: We seek the number N of independent properties needed to specify a state with P coexisting phases, with C total components present. We know the total amount of each of the C components in the system. Recall that for a pure phase of one component, three properties were su ffi cient to specify the phase ( N and V and E , for example, or N and T and P ). If the phase could have C di ff erent components, we would need C + 2 properties to specify the phase (one N for every component). So to specify P unrelated phases, we would need P ( C + 2) properties. But the phases are not unrelated. To start with, we ourselves specify the total amount of each of the C components, which is C constraints on the properties. And, there are equilibrium conditions that relate the phases. Between two phases in equilibrium, there are conditions specifying that the phases are in equilibrium with respect to the following “virtual exchanges”: Exchange of energy Exchange of volume Exchange of molecules of each type Thus there are ( C + 2) constraints imposed when two phases are in equilibrium. (Simply, these are that T and P must be equal in the two phases, in addition to C constraints of “equilibrium with respect to exchange of molecules between the phases” that we will discuss later in the course.) When P phases coexist, it is su ffi cient to impose P - 1 sets of such constraints, to relate all the phases. ((sketch of P - 1 relations between phases)) Thus the total number of constraints is C + ( P - 1)( C + 2). If we subtract this number from the total number of properties required to specify P phases, we get the number of unconstrained degrees of freedom N : N = P ( C + 2) - ( C + ( P - 1)( C + 2)) = C + 2 - P This is the Gibbs phase rule. As N must be nonnegative, P C + 2. 10

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Anatomy of a PVT surface Consider a single pure component. The Gibbs phase rule tells us that two intensive properties are su ffi cient to determine any other intensive property for a single phase. So if we consider the set of properties P, T, v (molar volume), specifying any two for a single phase is su ffi cient to determine the third. This means there is a surface of possible states, which we can plot in a three-dimensional plot.
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