# Binary Vapor Liquid Equilibrium (VLE)

This article shows how to prepare Pxy and Txy diagram for binary mixtures in excel spreadsheet based on Wilson, NRTL and UNIQUAC activity coefficient model.

For low to moderate pressure vapor liquid equilibrium (VLE) is described by modified Raoult’s Law –

`y`

= x_{i}P_{i}γ_{i}P_{i}^{sat}

where, y_{i} is vapor mol fraction, P is system pressure, x_{i} is liquid mol fraction, γ_{i} is activity coefficient and P_{i}^{sat} is vapor pressure for a pure component i. Vapor pressure is calculated based on Antoine equation.

`ln P`_{i}^{sat} = A_{i} - B_{i} /( T + C_{i} )

A_{i} , B_{i} and C_{i} are Antoine equation constants and T is temperature at which vapor pressure is to be calculated.

### Txy Diagram

Txy diagram plots bubble and dew point curves at constant pressure P. Put down the liquid mol fraction x_{1} from 0.0 to 1.0 with increment of 0.01 in spreadsheet. Iteration is done for each liquid mol fraction to estimate equilibrium temperature T and activity coefficient γ_{i}.

For first iteration, T_{1}^{sat} and T_{2}^{sat} are calculated from Antoine equation.

`T`_{i}^{sat} = B_{i}/ (A_{i} - ln P_{i}^{sat}) - C_{i}

Equilibrium temperature is estimated as following –

`T = x`_{1} T_{1}^{sat} + (1 - x_{1})T_{2}^{sat}

Based on temperature T, activity coefficient γ_{1} and γ_{2} are calculated from activity coefficient model selected e.g. Wilson, NRTL and UNIQUAC. For ideal mixture γ_{1 }and γ_{2} are 1.

Saturation pressure for a component is calculated using following equation –

`P`_{1}^{sat} = P/(x_{1}γ_{1} +(1-x_{1})γ_{2} P_{2}^{sat}/P_{1}^{sat})

Temperature corresponding to the vapor pressure P_{1}^{sat} is calculated from Antoine equation.

`T = B`_{1}/ (A_{1} - ln P_{1}^{sat}) - C_{1}

Temperatue thus calculated is used for next iteration and activity coefficients γ_{1} and γ_{2} are calculated. Iterations are repeated till there is no change in subsequent temperature estimations. Typically temperature difference becomes negligible within 10 iterations.

Above steps are repeated for all liquid mol fractions, thereby giving a table of x_{1} and corresponding temperature T. Vapor mol fraction y_{1} is calculated as following –

`y`_{1} = x_{1} γ_{1} P_{1}^{sat}/ P

Plot of T, x_{1} & y_{1} gives Txy Diagram –

### Pxy Diagram

Pxy diagram plots bubble and dew point curves at constant temperature T. Put down the liquid mol fraction x_{1} from 0.0 to 1.0 with increment of 0.01 in spreadsheet. Calculate activity coefficients γ_{1} and γ_{2} based on activity coefficient model selected from Wilson, NRTL and UNIQUAC.

Calculate partial pressure of each component P_{1} and P_{2} as following –

`P`

_{1}= x_{1}γ_{1}P_{1}^{sat}`P`

_{2}= (1 - x_{1}) γ_{2}P_{2}^{sat}

Equilibrium pressure is obtained as following –

`P = P`_{1} + P_{2}

Vapor mol fraction is calculated as per below equation.

`y`_{1} = P_{1} / P

Plot of P, x_{1} & y_{1} gives Pxy Diagram –

### Wilson Model

Activity coefficient for binary system are defined as –

Wilson parameter is provided by following equation –

where, λ_{12} – λ_{11} and λ_{21} – λ_{22} are binary interaction parameters available from literature for a binary pair.

Modified Rackett equation is used to estimate liquid molar volume V_{1} & V_{2}.

`V = (RT`_{c}/P_{c})Z_{RA} ^{[1 + (1-Tr)^(2/7)]}

where, T_{c} and P_{c} are critical temperature and pressure. T_{r} is the reduced temperature. Z_{RA} is Rackett equation parameter, if it is not available, it can be estimated from accentric factor ω as following.

`Z`_{RA} = 0.29056 - 0.08775ω

### NRTL Model

Activity coefficient for binary system are defined as –

Parameter g_{12} – g_{22} and g_{21} – g_{11} are binary parameters available from literature. α_{12} is related to non-randomness in mixture and is available from literature for binary pairs.

### UNIQUAC Model

Activity coefficient for binary system are defined as –

Parameter u_{12} – u_{22} and u_{21} – u_{11} are binary parameters available from literature. Remaining parameters are calculated as following –

where z is set equal to 10 and r, q & q’ are pure component UNIQUAC parameters.

### Spreadsheet

All above calculations along with iterative procedure for Txy diagram have been modeled in below spreadsheet. Sheets can be modified and more binary pairs can be added in data-bank.

## 3 Replies to “Binary Vapor Liquid Equilibrium (VLE)”

It is really a great post and pretty useful for chemical engineering students as well. Ten thumps up!!

Great work. I loved the spreadsheet very much.

just a question : i have a water-ethylene system and i checked the data bank and i didn’t find the αij and gij – gjj for ethylene water . where can i find them ?

the data bank is great i think you should upload them alone and if you make for raoult’s law

oh i forgot. later on on the process i have a water-ethanol-ethylene system . the activity coefficient will differ than the one above right ? how to obtain it .

Thanks again for your impressive work