# 17.2: Equilibrium and Equilibrium Ratios (Ki)

Consider a liquid-vapor in equilibrium. As we have discussed previously, a condition for equilibrium is that the chemical potential of each component in both phases are equal, thus:

(17.1)

We showed that this is equivalent to:

(17.2)

This is, for a system to be in equilibrium, the fugacity of each component in each of the phases must be equal as well. The fugacity of a component in a mixture can be expressed in terms of the fugacity coefficient. Therefore, the fugacity of a component in either phase can be written as:

(17.3a) (17.3b)

Introducing (17.3) into (17.2),

(17.4)

This equilibrium condition can be written in terms of the equilibrium ratio K_{i} = y_{i}/x_{i}, to get:

(17.5)

Do you recall the problem at the end of Module 13? At that point we needed a more reliable way to calculate the equilibrium ratios that showed up in the Rachford-Rice objective function. We demonstrated that once we know all values of K_{i}’s, the problem of vapor-liquid equilibrium is reduced to solving the Rachford-Rice objective function, using the Newton-Raphson Procedure.

We can now calculate equilibrium ratios, using (17.5), in terms of fugacity coefficients. We also know that we have an analytic expression for the calculation of fugacity coefficients via EOS — this was shown in the last section of the previous module. This is why we call this module “Vapor Liquid Equilibrium via EOS.”

Is this the end to our problems? Not quite. Take a look at the expression for fugacity coefficients in mixtures both for SRK EOS and PR EOS. It is clear that they are functions of the pressure, temperature, and composition of the phases:

(17.6a) (17.6b)

Do we know the composition of the phases “x_{i}”, “y_{i}” in advance? In a typical flash problem, we are given pressure, temperature and *overall* composition (z_{i}). What do we want to know? How much gas, how much liquid, and the *compositions of the phases*: α_{g}, α_{l}, y_{i}, x_{i}. So, we do *not* know those compositions in advance; therefore, as it stands, we cannot calculate (17.6) or (17.5). Thus far, it seems that the flash problem is unsolvable.

If we are bold enough, we could try to overcome this problem by “guessing” those compositions, and proceed by solving (17.6) and (17.5). With this “rough” estimate for Ki’s, we could solve for “α_{g}” with the procedure outlined in Module 13 (“Objective Function and Newton-Raphson Procedure”). Once “α_{g}” is known, we could back calculate the compositions of the phases using equations (12.7) and (12.11). If we were correct, those compositions would match each other (the “guessed” ones with respect to the “back-calculated”). More than likely, this would not happen, and we would have to make a new “guess.” This is, fundamentally, an iterative procedure. Although this is not what we do, it does illustrate that this problem is solvable by implementing the appropriate iterative scheme.

In equations (17.4) and (17.5), the fugacity of the liquid and vapor phases were computed in terms of the fugacity coefficient. Hence, this method of expressing the equilibrium criteria is known as the *dual-fugacity coefficient method.* For the sake of completeness, it is necessary to indicate that the fugacity of a component in a mixture can also be expressed in terms of a thermodynamic concept called the *activity coefficient*. While the fugacity coefficient is seen as a measure of the deviation of behavior with respect to the ideal gas model, the activity coefficient measures the deviation of behavior with respect to the ideal *liquid* model. This approach is called the *dual-activity coefficient method*, in which both liquid and vapor phase fugacities are expressed in terms of the activity coefficient and substituted into the equilibrium criteria in (17.2). A mixed *activity coefficient-fugacity coefficient method* can be also devised by expressing the liquid phase fugacities in terms of activity coefficients and the vapor phase fugacities in terms of fugacity coefficients. Each of the aforementioned methods for the calculation of phase equilibria has its advantages and disadvantages. The dual-fugacity-coefficient method is simpler both conceptually and computationally, but if the equation of state does not predict liquid and vapor densities well, the results may be inaccurate. The activity coefficient method can be more accurate, but it is more complicated to implement. For the rest of the discussion, the dual-fugacity coefficient approach will be used.

### Contributors

Prof. Michael Adewumi (The Pennsylvania State University). Some or all of the content of this module was taken from Penn State's College of Earth and Mineral Sciences' OER Initiative.