Non-equilibrium Thermodynamics of Heterogeneous Systems. Signe Kjelstrup
The following combination is frequently used in expression (4.15) for the entropy production and flux equations that derive from this:
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(3.62) |
Alternative expressions are as follows:
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(3.63) |
The partial molar surface area, the partial molar entropy and the partial molar polarization for the surface are, respectively:
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(3.64) |
Furthermore, we have
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(3.65) |
By using the partial molar quantities defined above, one may also define the partial molar internal energy and enthalpy
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(3.66) |
where these functions, according to the above construction, are functions of γ, Ts, γk and (Dseq/ε0).
Exercise 3.A.1.Find a form of Gibbs–Duhem’s equation for the surface that contains dμsj, T instead of dμjs.
•Solution: If we substitute Eq. (3.62) into Eq. (3.32d), we have
By using
3.A.3Standard states
The internal energy or the various derivatives of this energy, including the partial molar energies cannot be measured, only energy differences are measurable. In order to establish a measuring scale, we introduce the standard state as a point of reference. The standard state may be chosen freely, and different conventions have been made. Common to all choices is that they can be derived from each other by well-defined measurements.
The standard state for gases used in the SI system is the state of an ideal gas at 1 bar and constant temperature. The temperature is not specified in the definition of the standard state. Standard state values are often tabulated at 298 K, however. The definition of the chemical potential then relates any state to the standard state via
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(3.67) |
where Vm is the molar volume. For an ideal gas, we obtain
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(3.68) |
where μ0 is the standard chemical potential and p0 = 1 bar. The energy of a real gas is measured with respect to this standard state, with the fugacity f replacing the pressure p of the gas. The chemical potential is as follows:
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(3.69) |
where the fugacity coefficient, defined by the ratio ϕ = f/p, measures the deviation of the real gas from the ideal state, see Fig. 3.5 and Exercise 3.4. The chemical potential of ideal and real gases are illustrated in this figure. When μ − μid < 0 and ϕ < 1, attractive forces are important.
For a liquid solution, we use a liquid state as the standard state. The numbers to be compared become then closer to each other. Two choices are common; the Raoultian standard state (used for solvents) and the Henrian standard state (used for solutes). The Raoultian standard state is the state of a solution which obeys Raoult’s law. In this law, the vapor pressure above the solution is proportional to the mole fraction of the solvent (component 1):
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(3.70) |
where p1∗ is the vapor pressure above the pure solvent. We assume that the vapor above the solution is an ideal gas. Equation (3.68) gives then:
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(3.71) |
Figure 3.5The chemical potential of an ideal gas (stipled line) and a real gas (whole line). The standard state, μ0, at p0 = 1 bar is shown for the ideal gas. The deviation of the chemical potential of the real gas from the corresponding ideal gas value at p is measured by the term RT lnϕ.
which can be rearranged into
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(3.72) |
The Raoultian standard state for the solvent is defined by
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(3.73) |
The Raoultian standard state is thus determined from the standard state of the ideal gas plus a term that contains the vapor pressure of the solvent. A deviation from the Raoultian state is expressed by
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(3.74) |
where y1 is the activity coefficient that measures deviation from ideal behavior, and ai is the activity, a1 = p1/p1∗. The situation with y1 > 0 is illustrated in Fig. 3.6. Such a value means that there are repulsive forces