



Goal:
The composition of a binary solution with lowest
Gibbs free energy of mixing corresponds to the azeotrope. Gas chromatography data is used to calculate
the Gibbs free energy of mixing for various
cyclohexane and ethanol solutions. Based
on the uncertainty of all the measurements and using error propagation
techniques, the property is calculated with its associated uncertainty. This exercise is adequate for the
undergraduate physical chemistry curriculum. Prerequisites: Introductory thermodynamics, phase behavior. Resources you will need: EXCEL or similar software package Background: The molar Gibbs free energy of mixing (∆G_{mix}) for a binary solution is given by: ∆G_{mix }= RT (x_{1} ln a_{1} + x_{2} ln a_{2}) = RT (x_{1} ln g_{1} x_{1} + x_{2} ln g_{2} x_{2}) = RT (x_{1} ln g_{1} + x_{2} ln g_{2}) + RT (x_{1} ln x_{1} + x_{2} ln x_{2}) [1] where R is the ideal gas constant, T the absolute temperature, x_{i} the molar fraction of the i^{th} components, a_{i} its activity, and g_{i} its activity coefficient. The ideal molar Gibbs free energy of mixing (∆G_{mix}^{ideal}) and the excess molar Gibbs free energy of mixing (∆G_{mix}^{excess}) are then defined as: ∆G_{mix}^{ideal }= RT (x_{1} ln x_{1} + x_{2} ln x_{2}) [2] ∆G_{mix}^{excess} = RT (x_{1} ln g_{1} + x_{2} ln g_{2}) [3] The regular solution model assumes that the molar entropy of mixing ∆S_{mix} corresponds to an ideal solution, assumption that is reasonable for small molecules of similar size: ∆S_{mix }=  R (x_{1} ln x_{1} + x_{2} ln x_{2}). [4] Since: ∆G_{mix }= ∆H_{mix}  T ∆S_{mix} [5] then, the molar heat of mixing ∆H_{mix} for a real solution is given by: ∆H_{mix }= RT (x_{1} ln g_{1} + x_{2} ln g_{2}) = ∆G_{mix}^{excess} [6] The composition with the lowest Gibbs free energy of mixing (see equation [1]) corresponds to the azeotrope. The mole fractions (x_{i}) can be calculated from the composition of the binary solution. The activity coefficients (g_{i}) can be derived from an analysis of the composition of the distillate of the binary solution, using Raoult's and Considering the vapor pressure of the i^{th} component above the binary solution (P_{i}), Raoult's law convention states that: P_{i} = a_{i} P^{0}_{i }= g_{i} x_{i} P^{0}_{i}. [7] where P^{0}_{i} is the vapor pressure of the i^{th} pure component. If the vapor phase above the solution can be assumed to be ideal, P_{i} = y_{i} P_{total} [8] where y_{i} is the mole fraction for the i^{th} component in the vapor phase and P_{total} is the total pressure. If P_{total}, P^{0}_{i}, x_{i} and y_{i} are known, then the activity coefficients in the liquid phase (g_{i}) can be calculated by equating [7] and [8]: y_{i} P_{total} = g_{i} x_{i} P^{0}_{i} [9] Experimental Data: A series of binary solutions of cyclohexane and ethanol were prepared.
The solutions were analyzed with a gas chromatograph (GC) and the peak heights were recorded (h_{i}) for the two components.
The solutions were boiled, the distillates were collected and analyzed using the GC under the same conditions, and the peak heights were recorded (h'_{i}) for the two components.
In order to calculate the molar fraction in the gas phase (y_{i}), standard curves are constructed for the two components of the binary solutions, using mole fractions (x_{i}) and peak height functions (H_{i}) from the liquid phase.
The barometric pressure and the room temperature were measured. The reported value of barometric pressure includes a temperature correction.
The molar Gibbs free energy of mixing (∆G_{mix}) for the various solutions is calculated from equation [1]. Other thermodynamic terms (∆G_{mix}^{ideal}, ∆G_{mix} ^{excess}, ∆S_{mix}, ∆H_{mix}) are also obtained. All values are calculated with their associated uncertainty, based on the uncertainty in the data. Experimental Conditions  further information that will be needed:
Composition of the solutions^{*7}:
GC peak heights for the distillates (cm):
Exercise: (Note: See Sample Calculations below) 1. Calculate the mole fractions for the solutions (x_{i}), and their uncertainties. Remember that the error of sums or subtractions is the sum of the errors of the individual terms and that the relative error of products or divisions is the sum of the relative errors of the individual terms. 2. Calculate the peak height functions for the solutions (H_{i}), as well as for the distillates (H'_{i}). 3. Obtain thirdorder polynomial fits of x_{i} vs H_{i} for cyclohexane and for ethanol. 4. Using the thirdorder polynomial functions, calculate the mole fractions in the distillates (y_{cy} and y_{et}) as a function of the peak height function of the distillates (H'_{cy} and H'_{et}). Assume that the associated uncertainty for the y values is the standard error of the y estimates. 5. Using the corresponding Antoine equation, calculate the vapor pressure of pure cyclohexane and ethanol (P^{0}_{i}) at the boiling temperatures. Calculate the uncertainty for the vapor pressures of the pure compounds, at the boiling temperatures, using the formulas: .
8. Calculate ∆H_{mix} (or
∆G_{mix}^{excess}) and ∆S_{mix} (or ∆G_{mix}^{ideal}/T)
and estimate their associated errors.9. Plot ∆G_{mix} vs x_{i}. Fit the data into a thirdorder polynomial function. Calculate the azeotrope composition by finding the value of x_{i} at the minimum of the ∆G_{mix }curve. 10. Estimate, by interpolation, the boiling temperature for the azeotrope. Questions: 1. What assumption is made concerning the vapor phase produced at the boiling point? Is it valid? Sample Calculations: 1. Mole fraction for cyclohexane (x_{cy}) in solution 2: Associated uncertainty for x_{cy} in solution 2 (∆x_{cy}): 2. Peak height functions for cyclohexane in solution 2 (H_{cy} and H'_{cy}):
Using LINEST(known_y's,[known_x's],true,true) {CtrlShift,Enter}
Coefficient of determination = 0.9996 Standard error of the y estimates: 0.0095. 4. Mole fraction for cyclohexane in the distillate of solution 2: y_{cy} = a + b H'_{cy} + c H'_{cy}^{2}
+ d H'_{cy}^{3}
Dy_{cy} = 0.0095
5. Vapor pressure for pure
cyclohexane in solution 2:6. Activity coefficient for cyclohexane in solution 2: 7. Molar Gibbs free energy of mixing (∆G_{mix}) for solution 2; Associated uncertainty in solution 2 [∆(∆G_{mix})]: 8. Other thermodynamic properties:
Similarly to ∆G_{mix}, selecting
only certain terms.
9. Location of the azeotrope:
Use LINEST to obtain the polynomial fit
∆G_{mix} = a +
b x_{cy} + c x_{cy}^{2} + d x_{cy}^{3}.
Minimize
the function with respect to x_{cy}.
(d ∆G_{mix / }d x_{cy}) = b + (2 c) x_{cy,min} + (3 d) x_{cy,min}^{2}
= 0
Solve for x_{cy,min}.
10. Given that x_{cy,min} is between x_{cy,j} and x_{cy,k}, with boiling temperatures T_{j} and T_{k}:
Solve for T_{min}.
Notes and references: ^{*1 }Includes a barometer temperature correction of 2.7 torr based on Lange's Handbook of Chemistry, Editor John A. Dean, Copyright 1973 by McGrawHill, Inc., page 230. ^{*2} http://organicdivision.org/organic_solvents.html; last accessed ^{*3} Based on data from B.E. Poling, J. M. Prausnitz, J. P. O'Connell, "The Properties of Gases and Liquids", 5th Edition, McGrawHill, 2001, fitted between 60^{o} and 85^{o}C. Pressure in torr; temperature in K. ^{*4} Based on data from R.C. Reid, J. M. Prausnitz, B. E. Poling, "The Properties of Gases and Liquids", 4th Edition, McGrawHill, 1987, fitted between 60^{o} and 85^{o}C. Pressure in torr; temperature in K. *^{5} The uncertainty is estimated from the unreported decimal place. *^{6} kdb/hcprop/cmpsrch.php; last accessed ^{*7} Experiment performed ^{*8} Accuracy of pipettes: 5mL = 1%; 10mL = 0.8%; in 20mL = 0.5%. *^{9} CRC Handbook of 

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