Boiling point diagram of a two-component mixture

Theory

The aim of this lab course is to obtain knowledge on the boiling behavior of a mixture of two components. This knowledge is crucial for the distillation purification of mixtures.

Phase equilibrium of an ideal two-component mixture

As a simple theoretical description we consider first the ideal mixture, where we neglect any interactions between the components of the mixture.

Gibbs phase rule

The Gibbs phase rule describes how many parameters of a system can be changed independently without changing the number of phases present in the system . We denote the number of independently changeable parameters (degrees of freedom) with $F$ , the number of phases in the system with $P$ and the number of components with $C$ . The Gibbs phase rule then states that:

F=C-P+2

If a system consists of both a liquid and a gas phase in thermodynamic equilibrium we call the system boiling. The two phases of the system we consider here are a mixture of two different components. According to the Gibbs phase rule a boiling two-component system has two degrees of freedom. By stating the pressure, temperature and composition of each phase we can fully describe the system.

Thermodynamic equilibrium

In thermodynamic equilibrium the following aspects have to be considered:

The temperature is equal in all phases of the system. The temperature of a system where both liquid and gas phase coexist is called boiling temperature $T_\text b$ .
The pressure is equal in all phases of the system. The pressure is denoted by $p$ .
For each component, the chemical potential in both phases is equal.

From the Gibbs phase rule it follows that the phase interface liquid $\leftrightarrow$ gas in a phase diagram corresponds to an area. We speak of a two-phase region to emphasize the difference with respect to the vapor pressure curve of a one-component system.

A state in the two-phase region is uniquely determined by stating three parameters, i.e. pressure, boiling temperature and the composition of one of the phases.

Composition and pressure

In this section we prepare equations needed later to describe the relationship between the composition and the pressure. We consider a mixture of two pure substances A and B.

To quantify the composition of the liquid phase we use the mole fractions $x_\text A$ and $x_\text B$ as the quotient of the amount of substance of each component with the total amount of substance in the liquid phase. It follows that

x_\text A+ x_\text B=1.

Similarly, to quantify the composition of the gas phase we use the mole fractions $y_\text A$ and $y_\text B$ as the quotient of the amount of substance of each component with the total amount of substance in the gas phase. It follows that

y_\text A+ y_\text B=1.

When assuming an ideal behavior of the gas phase, the amount of substance in the gas phase is proportional to the partial pressure $p_\text A$ and $p_\text B$ of each component, respectively. The vapor pressure of a mixture $p$ is the sum of the partial pressure of the components.

y_\text A = \frac{p_\text A}{p_\text A + p_\text B}=\frac{p_\text A}{p}

y_\text B = \frac{p_\text B}{p_\text A + p_\text B}=\frac{p_\text B}{p}

Boiling point diagram

Now we assume that at the temperature $T_\text{b,A}^\text{pure,ref}$ and at the pressure $p_\text{A}^\text{pure,ref}$ the pure substance A boils (figure ).

Schematic depiction of the two phases of the boiling pure substance A.

Similarly, at the temperature $T_\text{b,B}^\text{pure,ref}$ and the pressure $p_\text{B}^\text{pure,ref}$ the pure substance B boils. Using the August vapor pressure equation we can express the vapor pressure for pure substance A $p_\text{A}^\text{pure}$ at arbitrary boiling temperatures $T_\text{b,A}^\text{pure}$ .

p^\text{pure}_\text A\left(T_\text{b,A}^\text{pure}\right) = p_\text{A}^\text{pure,ref}\exp{\left(\frac{\Delta_\text{vap} H_\text{m,A}^\text{pure}}{R}\left(\frac{1}{T_\text{b,A}^\text{pure,ref}}-\frac{1}{T_\text{b,A}^\text{pure}}\right)\right)}

Here $R$ is the universal gas constant $R=8.314\,\text J / (\text{mol}\,\text K)$ and $\Delta_\text{vap}H_\text{m,A}$ is the molar evaporation enthalpy of the pure substance A. The derivation of the August vapor pressure can be found in the instructions for the lab course vapor pressure. Similar, the vapor pressure $p_\text{B}^\text{pure}$ at the boiling temperature $T_\text{b,B}^\text{pure}$ is described as

p^\text{pure}_\text B\left(T_\text{b,B}^\text{pure}\right) = p_\text{B}^\text{pure,ref}\exp{\left(\frac{\Delta_\text{vap} H_\text{m,B}^\text{pure}}{R}\left(\frac{1}{T_\text{b,B}^\text{pure,ref}}-\frac{1}{T_\text{b,B}^\text{pure}}\right)\right)}

Next, we consider boiling a mixture of the substances A and B. The mixture boils at a temperature of $T_\text{b}$ and a vapor pressure $p$ , which depend on the composition of the mixture (see figure ).

Schematic depiction of the two phases of the boiling mixture.

In the ideal case, the partial pressure of each pure substance in the gas phase of the boiling mixture would correspond to the weighted vapor pressure of the pure substance at the same boiling temperature. The resulting expression $p_\text A=x_\text A p_\text A^\text{pure}$ is called Raoult law. The vapor pressure of the mixture $p$ then follows as the sum of the partial pressures of the components.

p = p_\text A + p_\text B = x_\text A p_\text A^\text{pure} + x_\text B p_\text B^\text{pure}

Rearranging the Raoult law using equations , , and yields

\frac{y_\text A}{x_\text A}=\frac{p_\text A^\text{pure}}{p}\quad\text{and}\quad \frac{1-y_\text A}{1-x_\text A}=\frac{p_\text B^\text{pure}}{p}.

In this lab course, the boiling temperatures of the pure substances $T_\text{b,A}^\text{pure,ref}$ at $p_\text{A}^\text{pure,ref}$ and $T_\text{b,B}^\text{pure,ref}$ at $p_\text{B}^\text{pure,ref}$ are recorded at the same environmental pressure $p$ as the boiling temperature of the mixture $T_\text{b}$ . The temperature of the mixture is constant in the two phases.

\begin{aligned} p_\text{A}^\text{pure,ref}=p_\text{B}^\text{pure,ref}=p_\text{\vphantom{A}}^\text{\vphantom{pure,ref}} \\ T_\text{b,A}^\text{pure}=T_\text{b,B}^\text{pure}=T_\text{b}^\text{\vphantom{pure}} \end{aligned}

From equation , the August vapor pressure equation, equation and it follows that

\frac{y_\text A}{x_\text A}=\exp{\left(\frac{\Delta_\text{vap} H_\text{m,A}^\text{pure}}{R}\left(\frac{1}{T_\text{b,A}^\text{pure,ref}}-\frac{1}{T_\text{b}^\text{\vphantom{pure,ref}}}\right)\right)}

\frac{1-y_\text A}{1-x_\text A}=\exp{\left(\frac{\Delta_\text{vap} H_\text{m,B}^\text{pure}}{R}\left(\frac{1}{T_\text{b,B}^\text{pure,ref}}-\frac{1}{T_\text{b}^\text{\vphantom{pure,ref}}}\right)\right)}

If the boiling temperature of the pure substances are similar to the boiling temperature $T_\text{b}$ of the mixture, the exponents in equation and equation are small and the exponential function can be approximated as a Taylor series up to the linear term (with $\exp{a}=1+a$ ).

This allows to resolve both equations and for the boiling temperature of the mixture $T_\text{b}$ as function of the composition of the liquid phase $x_\text A$ . After some rearrangement we obtain $T_\text{b}$ as follows:

T_\text{b}=T_\text{b,B}^\text{pure,ref}\frac{1+x_\text A\left(\frac{\Delta_\text{vap}H_\text{m,A}^\text{pure}}{\Delta_\text{vap}H_\text{m,B}^\text{pure}}-1\right)}{1+x_\text A\left(\frac{\Delta_\text{vap}H_\text{m,A}^\text{pure}}{\Delta_\text{vap}H_\text{m,B}^\text{pure}}\frac{T_\text{b,B}^\text{pure,ref}}{T_\text{b,A}^\text{pure,ref}}-1\right)}

A graphical plot of the boiling temperature $T_\text{b}$ of the mixture against the composition of the liquid phase $x_\text A$ and against the composition of the gas phase $y_\text A$ is called isobaric boiling point diagram.

Real two-component mixtures

Often various mixtures do not behave according to the idealized discussion above, since interactions between the various components (A $\leftrightarrow$ B) are different from the interactions with molecules of the same kind (A $\leftrightarrow$ A, B $\leftrightarrow$ B). Therefore, actual boiling point diagrams can exhibit a very different shape compared to the ideal one.

If the composition of the gas phase is equal to the composition of the liquid phase the boiling system is called azeotrope. The azeotrope is characterized by stating the azeotropic composition, the azeotropic boiling temperature and the azeotropic pressure. The boiling temperature of the azeotropic mixture is always at an extremum. At this point, it is not possible to purify the mixture by distillation.

We will employ a simple model to consider interactions between the components A and B of the mixture in the liquid phase.

If the interactions in the liquid phase between component A and B are less than those between the pure substances e (A $\leftrightarrow$ A, B $\leftrightarrow$ B), the azeotropic boiling temperature is at a minimum.
If the interactions in the liquid phase between component A and B are larger than those of the respective pure substances (A $\leftrightarrow$ A, B $\leftrightarrow$ B), the azeotropic boiling temperature is at a maximum.

In the scope of this lab course we do not consider other models that consider azeotropic mixtures in more detail.

Experimental determination of the mixture composition

The index of refraction $n_\text {A}$ and $n_\text {B}$ of the liquid pure substances A and B, respectively, is a characteristic property of each pure substance. The index of refraction $n$ of a mixture of A and B interpolates approximately linearly on the index of refraction of the pure substances.

n= x_\text A n_\text {A} + x_\text{B} n_\text{B}

Experimental setup

Distillation apparatus

The distillation apparatus consist of a vessel with a tempering jacket and a distillation bridge. Using a thermostat the tempering jacket can be heated to bring the mixture to a boil. The boiling temperature can be read from a temperature probe. The gas phase rises to a cooling finger where it condenses. The condensate drops off of the cooling finger into a depression with a tap (called steam tap in the following). Using this steam tap the condensate can be transferred into a flask. The liquid phase can be transferred into a flask using a tap (called liquid tap in the following) as well.

Abbe refractometer

The purpose of the Abbe refractometer is to determine the index of refraction of liquids through determination of their angle of total reflection. The index of refraction of a liquid depends on temperature and the wavelength of the used light source. For this reason, both quantities are usually stated in addition to the refractive index. Since all refractive indices in this lab course are taken at a wavelength of 589 nm and a temperature of 298.15 K, we denote the index of refraction simply by $n$ .

Instructions

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Lab course instructions

Preparation

Start the thermostat for the refractometer and the thermostat for the distillation apparatus and start the cooling water circulation (for both the refractometer and the thermostat).
Check using the flow sensor whether the cooling circulation is actually active.
Set the thermostat for the distillation apparatus to a Celsius temperature of 80 ℃.
Set the thermostat for the refractometer to a Celsius temperature of 25 ℃.
Measure the pressure in the closed fume hood. Notice the fluctuations of the pressure during the course of the experiment.

Notes

Make sure to fix all flask stoppers with a clip.
Position and glassware in such that residual contents can evaporate.
Determine the index of refraction with four decimal places.

Measurement of the boiling temperatures of the pure substances

Determine the boiling temperature and the index of refraction of pure cyclohexane and 2-propanol (corresponding to the schema in figure ).

Close all taps of the distillation apparatus.
Remove the glass stopper of the apparatus and replace it with a funnel.
Fill the apparatus slowly with approximately 70 mL of one of the pure substances over the funnel using the measuring cylinder.
Make sure that the temperature probe is inserted into the liquid (add more if necessary).
Replace the funnel with a glass stopper.
Set the thermostat for the apparatus to a Celsius temperature of 95 ℃.
Wait until approximately five drops of condensate accumulate at the steam tap. Note down the boiling temperature at this time.
Switch of the thermostat for the apparatus.
Remove the liquid by placing the corresponding container below the steam tap (and, afterwards, the liquid tap).
Remove the glass stopper from the apparatus to allow the remaining pure substance to evaporate.

Repeat the above steps for the second pure substance. Determine the index of refraction of both pure substances each three times using the refractometer. Clean the refractometer after each measurement and readjust.

Measurement of the boiling temperature and composition of the mixtures

Determine the boiling temperature and the index of refraction of various mixtures of cyclohexane and 2-propanol (corresponding to the scheme in figure ).

Close all taps of the apparatus.
Attach a 10 mL flask below the cooling finger.
Attach a 50 mL flask below the liquid reservoir.
Exchange the glass stopper of the apparatus with a funnel.
Fill the apparatus slowly with approximately 70 mL of one of a mixture over the funnel using the measuring cylinder.
Make sure that the temperature probe is inserted into the liquid (add more if necessary).
Replace the funnel with a glass stopper.
Set the thermostat for the apparatus to a Celsius temperature of 95 ℃.
Perform the next steps quickly: Wait, until five drops of condensate have formed at the gas tap. Note the boiling temperature.
Open - preferably at the same time - the steam tap and the liquid tap. Close the steam tap directly after. Close the liquid tap until a similar amount of liquid has been collected as at the steam tap.
Remove the lower flask and close it directly with a glass stopper.
Turn off the left thermostat.
Remove the upper flask and determine the index of refraction $n^\text g$ of the gas condensate three times using the refractometer. Clean and readjust the refractometer after each measurement. Close the flask directly after taking a sample.
Remove the mixture from the apparatus by placing the corresponding container under the apparatus and opening the steam tap (and then the liquid tap). Keep both taps open.
Take the glass stopper off the apparatus such than any residual liquid can evaporate.
Determine the index of refraction $n^\text l$ of the liquid phase three times. Clean and readjust the refractometer after each measurement.

Repeat the above steps to measure all mixtures.

Switch off all devices and the cooling circulation.

Notes for the interactive experiment

Wait about 10 seconds until the Celsius boiling point has reached a constant value after selecting the substance. Record the measured value by clicking on Record data.

Analysis

The index CH is used in the following on quantities which refer to cyclohexane, the index IP on quantities referring to 2-propanol (isopropyl alcohol). Compute, where necessary, the temperature from the Celsius temperature.

Determination of the composition

Compute the average of the measured index of refraction. Determine the composition of the liquid phase $x_\text{CH}$ and of the gas phase $y_\text{CH}$ of each mixture using the average index of refraction $n_\text{IP}$ and $n_\text{CH}$ of the pure substances, the averaged index of refraction of the mixtures' liquid phase $n^\text l$ and gas phase $n^\text g$ , respectively. Use the rearranged form of equation for this purpose:

\begin{aligned} x_\text{CH} &= \frac{n^\text{l}-n_\text{IP}}{n_\text{CH}-n_\text{IP}}\\ y_\text{CH} &= \frac{n^\text{g}-n_\text{IP}}{n_\text{CH}-n_\text{IP}} \end{aligned}

Determination of the ideal boiling temperature

Calculate the ideal boiling temperature $T_\text{b}^\text{ideal}$ for each mixture as function of the composition of the liquid phase. Use your measured boiling temperature of the pure substances $T^\text{pure,ref}_\text{b,IP}$ and $T^\text{pure,ref}_\text{b,CH}$ . Do not perform any calculation of the error/uncertainty for $T_\text{b}^\text{ideal}$ and state the result with one decimal place.

T_\text{b}^\text{ideal}=T_\text{b,IP}^\text{pure,ref}\frac{1+x_\text{CH}\left(\frac{\Delta_\text{vap}H_\text{m,CH}^\text{pure}}{\Delta_\text{vap}H_\text{m,IP}^\text{pure}}-1\right)}{1+x_\text{CH}\left(\frac{\Delta_\text{vap}H_\text{m,CH}^\text{pure}}{\Delta_\text{vap}H_\text{m,IP}^\text{pure}}\frac{T_\text{b,IP}^\text{pure,ref}}{T_\text{b,CH}^\text{pure,ref}}-1\right)}

Calculate from the ideal boiling temperature with equation the composition of the gas phase. Since the quantity is computed from the ideal boiling temperature we call the obtained quantity ideal composition of the gas phase $y_\text{CH}^\text{ideal}$ .

y_\text{CH}^\text{ideal} = x_\text{CH}\exp{\left(\frac{\Delta_\text{vap}H_\text{m,CH}^\text{pure}}{R}\left(\frac{1}{T_\text{b,CH}^\text{pure,ref}}-\frac{1}{T_\text{b}^\text{ideal}}\right)\right)}

Do not perform any calculation of the error/uncertainty for $y_\text{CH}^\text{ideal}$ and state the result with two decimal places.

Use the molar enthalpy of evaporation

\begin{aligned} \Delta_\text{vap} H_\text{m,CH}^\text{pure} &= 30\, \text{kJ}\,/\,\text{mol}\\ \Delta_\text{vap} H_\text{m,IP}^\text{pure} &= 40\,\text{kJ}\,/\,\text{mol} \end{aligned}

Isobaric boiling point diagram

Create a isobaric boiling point diagram using the measured and ideal values in a single diagram.

Plot the boiling temperature $T_\text{b}$ of each mixture against the composition of the liquid phase $x_\text{CH}$ and the composition of the gas phase $y_\text{CH}$ . Also plot the boiling temperature of the pure substances.

In the same diagram, plot the ideal boiling temperature $T_\text{b}^\text{ideal}$ of each mixture against the composition of the liquid phase $x_\text{CH}$ and against the ideal composition of the gas phase $y_\text{CH}^\text{ideal}$ .

Connect for the purpose of better visibility corresponding points with a line.

Discuss the boiling point diagram and determine the composition and boiling temperature of the azeotropic mixture. Determine a reasonable error/uncertainty for the two quantities.

Compare your results with the following literature values :

\begin{aligned} p_\text{azeotrope}^\text{lit.} &= 1013 \text{hPa}\\ T_\text{b,azeotrope}^\text{lit.} &= 342{,}6 \text K\\ x_\text{CH,azeotrope}^\text{lit.} &= 0{,}603 \end{aligned}