2.1 Formulae for calculating activity coefficients using the ASOG method

2024.9.04

Chemistry at pirika.com > Chemistry > Chemical engineering > Reprint: gas-liquid equilibrium estimation by ASOG. > Chapter 1: Basic equations of solution theory > Chapter 2: ASOG method. >

2.1 Formulae for calculating activity coefficients using the ASOG method

The basic equations for determining the activity coefficient using the ASOG method are the following six.

ln γ_i = ln γ_i^FH + ln γ_i^G　　　 (2-1)
ln γ_i^FH = ln⁡ (v_i^FH ) / (∑_j (v_j^FH x_j ) + 1- (v_i^FH )/∑_j(v_j^FH x_j ) 　　 (2-2)
ln⁡ γ_i^G =∑_k v_ki (ln Γ_k – ln⁡ Γ_k⁽ⁱ⁾ ) 　　 (2-3)
ln⁡Γ_k = -ln∑_l X_l a_kl +1- ∑_l(X_l a_lk) / ∑_m(X_m a_lm ) 　　(2-4)
X_l=∑_i x_i v_li / ∑_ix_i ∑_kv_ki 　　　(2-5)
ln⁡ a_kl = m_kl + n_kl/T 　　(2-6)

γ_i：activity coefficient
ln γ_i^FH ：Contribution of different component molecules in solution.
lnγ_i^G ：Contributions showing inter-group interaction.
v_j^FH ：Number of atoms, excluding hydrogen atoms, in pure component j
x_j：Mole fraction of component j in solution
v_ki：Number of atoms in component i excluding hydrogen atoms in group k
Γ_k：Group activity coefficient for group k
Γ_k⁽ⁱ⁾：Group activity coefficient of k in the standard state (net component i)
a_kl：Group interaction parameters for groups k and l(a_kl ≠a_lk)
X_l：Group fraction of group l in solution

The formulae handled in chemical engineering make extensive use of superscripts, subscripts, Σ, etc., which makes it very difficult to understand what you are doing.
But don’t worry: the computer does the calculations for Σ, etc. automatically.

There are four values that have to be given to the computer: v_j^FH , v_ki, m_kl and n_kl.
These parameters are determined by choosing two (or more than two, the same) pure components.
Let’s look at these parameters in the first example ethanol/water system.

First, v_j^FH is the number of atoms in the pure component j, excluding hydrogen atoms. Ethanol is CH3CH2OH, so v_ethanol^FH is 3.
Water is H2O, so v_water^FH is 1.

In v_ki, put the number of atoms, excluding hydrogen atoms, in group k in component i. (However, CH3-, -CH2-, >CH- and >C< must be counted as 1:1:0.8:0.5 atoms.) Ethanol has no branches, so v_{(CH2,ethanol)}=2 and v_(OH,ethanol)=1 are counted. Water is also one group with one molecule, but counted as 1.6. v_(H2O,water)=1.6.

It is important to note that v_j^FH and v_ki are determined by one pure component. Therefore, if the values are registered in the solvent database, they can be identified simply by selecting the solvent.

Next, the group interaction parameter a_kl is obtained for groups k and l. The three groups present in this ethanol/water are CH2, OH and H2O. The group interaction parameter a_kl is an asymmetric (a_kl ≠ a_lk) value, so we have to find six values: a_(CH2,OH), a_(OH,CH2), a_(CH2,H2O), a_(H2O,CH2), a_(OH,H2O), a_(H2O,OH).

However, these values are also already determined as ASOG parameters. So, once the solvent pairs are determined, they are simply brought from the database.

In other words, to determine the activity coefficient γ_i, you only need to determine two solvents. Simple, isn’t it?

Next section: 2.2 Estimation of the gas-liquid equilibrium using the ASOG method

---

Copyright pirika.com since 1999-
Mail: yamahiroXpirika.com (Replace X with @.)
The subject line of the email should start with [pirika].