Vapor pressure - Physical Chemistry Lab Course

Theory

The aim of this lab course is to obtain information regarding the behavior of a boiling pure substance.

Phase equilibrium of one-component systems

We consider the thermodynamic equilibrium of a system that consists of one component only (a pure substance). We say that the system is boiling if both a liquid and a gaseous phase exist simultaneously. In thermal equilibrium, the pressure, temperature and chemical potential of this pure substance is equal in each phase. We call the temperature boiling temperature $T_\text b$ .

The chemical potential of the pure substance $\mu^\text l$ and $\mu^\text g$ in the liquid and the gaseous phase of the system is the molar free enthalpy $G_\text m^\text l$ and $G_\text m^\text g$ .

\begin{aligned} && \mu^\text{l} &= \mu^\text{g}\qquad\qquad\\ &\Leftrightarrow& G_\text m^\text{l} &= G_\text m^\text{g}\\ &\Leftrightarrow& \mathrm dG_\text m^\text{l} &= \mathrm dG_\text m^\text{g} \end{aligned}

The total differential of the molar free enthalpy is

\mathrm dG_\text m^\text{l} = V_\text m^\text l \mathrm dp - S_\text m^\text l \mathrm dT_\text b

\mathrm dG_\text m^\text{g} = V_\text m^\text g \mathrm dp - S_\text m^\text g \mathrm dT_\text b

Here $V_\text{m}^\text l$ and $V_\text{m}^\text g$ are the molar volume and $S_{\text m}^\text l$ as well as $S_{\text m}^\text g$ are the molar entropy of the liquid and gaseous phase.

From equation , and follows that in thermal equilibrium

\mathrm dp=\frac{S_\text m^\text g-S_\text m^\text l}{V_\text{m}^\text g-V_\text{m}^\text l} \mathrm dT_\text b.

The difference between the molar entropy of the liquid and the gaseous phase is called molar evaporation entropy $\Delta_\text{vap}S_\text{m}$ . The difference between the molar volume of the liquid and gaseous phase is called molar evaporation volume $\Delta_\text{vap}V_\text{m}$ .

Vapor pressure curve

From the definition of the molar free enthalpy $G_\text m = H_\text m-S_\text mT$ follows that

G_{\text{m}}^\text g = H_{\text{m}}^\text g -S_{\text{m}}^\text gT_\text b

\text{and}\qquad G_{\text{m}}^\text l = H_{\text{m}}^\text l -S_{\text{m}}^\text lT_\text b.

Since in thermal equilibrium the molar free enthalpy of each phase is equal, the difference between equations and yields

G_{\text{m}}^\text g-G_{\text{m}}^\text l = \underbrace{\left(H_{\text{m}}^\text g-H_{\text{m}}^\text l\right)}_{=\Delta_\text{vap}H_\text{m}} -\underbrace{\left(S_{\text{m}}^\text g-S_{\text{m}}^\text l\right)}_{=\Delta_\text{vap}S_\text{m}}T_\text b \stackrel{!}{=} 0

The difference between the molar enthalpy of the liquid and gaseous phase is called molar evaporation enthalpy $\Delta_\text{vap}H_\text{m}$ .

This allows to express the molar evaporation entropy using the molar evaporation enthalpy:

\Delta_\text{vap}S_\text{m}=\frac{\Delta_\text{vap}H_\text{m}}{T_\text b}

The molar volume of the gaseous phase is typically significantly larger than the molar volume of the liquid phase:

\Delta_\text{vap}V \approx V_\text{m}^\text g

The molar volume of the gaseous phase can then be expressed by using the ideal gas equation

V_\text{m}^\text g = \frac{RT_\text b}{p}

The universal gas constant is $R=8.314 \text J / (\text{mol} \text K)$ . Using equation , and in equation , we obtain the expression

\frac{1}{p}\mathrm dp = \frac{\Delta_\text{vap}H_\text{m}}{RT_\text b^2}\mathrm dT_\text b.

This differential equation can be solved by integration, if we approximate that the molar evaporation enthalpy does not depend on temperature. The lower integration bounds are set to a pressure $p^\text{ref}$ at which the pure substance boils at a temperature of $T_\text b^\text{ref}$ .

Interactive visualization of the August vapor pressure equation

\begin{aligned} \int_{p^\text{ref}}^{p}\frac{1}{p'}\mathrm dp' &= \int_{T^\text{ref}_\text b}^{T_\text b}\frac{\Delta_\text{vap}H_\text{m}}{RT_{\text b}'^2}\mathrm dT'_\text b\\ \Leftrightarrow\quad \ln{\left(\frac{p}{p^\text{ref}}\right)} &= \frac{\Delta_\text{vap}H_\text{m}}{R}\left(\frac{1}{T^\text{ref}_\text b}-\frac{1}{T^\text{\vphantom{ref}}_{\text b}}\right) \end{aligned}

Equation is called August vapor pressure equation. The equation describes the dependency of the vapor pressure $p$ of a pure substance as a function of temperature $T_\text b$ . A plot of the vapor pressure $p$ versus the boiling temperature $T_\text b$ is called vapor pressure curve.

Pictet-Trouton rule

The Pictet-Trouton rule states that the evaporation entropy of most liquids is approximately equal. . This was discovered around the year 1900 from the physicist Trouton, who noticed that the quotient of evaporation enthalpy and boiling temperature (equation ) approximates roughly 88 J / (mol K). Here, the boiling temperature refers to a pressure of 1013 hPa. Thus, it follows that

\Delta_\text{vap}S_\text{m}=88 \text J / (\text{mol} \text K).

The Pictet-Trouton rule states that the molecular order for a phase transition from liquid to gas changes by a similar amount. Large deviations of the Pictet-Trouton rule are typically observed for liquids consisting of polar molecules (that form, for example, hydrogen bonds).

Experimental setup

The experimental setup consists of a three-jacket vessel with a temperature sensor and a pressure sensor. The inner vessel contains the liquid pure substance and is surrounded by a tempering jacket. A constant temperature is achieved using a thermostat. A further vacuum jacket surrounds the temperature jacket to reduce condensation of water from air. A diaphragm pump is used to remove air from the gaseous phase within the setup and to reduce the pressure within the setup to the vapor pressure of the substance.

Instructions

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Lab course instructions

Preparation

Set the cryostat to a Celsius temperature of 15 ℃.
Check whether there is a stirrer in the three-jacket vessel.
Turn on the thermometer and the manometer.

Measurement

Vapor pressure of acetone

Fill the vessel up to the mark using the syringe with acetone.
Close the vessel using the thermal probe and switch on the magnetic stirrer.
When the temperature of the liquid is constant, the air in the gas phase of the vessel has to be replaced with the vapor of the liquid:
- Close both taps and turn on the pump.
- Slowly open the vapor valve. Close the vapor valve again as soon as the liquid inside the vessels boils and wait for 10 s.
- Repeat the above step three times.
- Close the vessel by firmly closing the vapor valve. Open the air valve and wait for one minute. This ensures that condensed liquid residues are removed from the pump. After this step, turn off the pump and leave the air valve open.
Start the measurement by setting the cryostat to a Celsius temperature to 35 ℃. Note down the pressure and the Celsius temperature in 10 s intervals. It is important to record both measurements at the same time while the exact time difference between the measurements in not important.
After recording the measurements the vessel is opened using the vapor valve and the liquid is transferred to the stock vessel using the syringe. Wait for approximately one minute to make sure that the acetone residue in the vessel evaporates completely.

Vapor pressure of ethanol

Set the cryostat to a Celsius temperature of 35 ℃.
Proceed analogously to the measurement of acetone. Make sure to use the right syringe. For starting the measurement, set the cryostat to a Celsius temperature of 55 ℃.
At the end of the measurement, transfer the liquid again to the stock vessel using the syringe.

Clean up

Switch off the lower and upper unit of the cryostat, the magnet stirrer, the thermometer and the manometer. Leave the vessel open to allow residual liquid to evaporate. Do not clean the vessel with water.

Analysis

Analyse your data for acetone and ethanol using the following steps:

Plot the measured pressure $p$ against the boiling temperature $T_\text{b}$ in a single diagram for both ethanol and acetone and discuss your data.
Plot according to equation the term $\ln{\left(\frac{p}{p^\text{ref}}\right)}$ against the inverse boiling temperature $\frac{1}{T_\text b}$ in a single diagram for both acetone and ethanol. Choose as reference pressure $p^\text{ref}$ the normal pressure of 1013.25 hPa. Determine the molar enthalpy of evaporation $\Delta_\text{vap} H_\text{m}$ and the boiling temperature $T_\text{b}^\text{ref}$ at normal pressure.
From the boiling temperature at normal pressure and the molar evaporation enthalpy determine according to equation the molar entropy of evaporation $\Delta_\text{vap} S_\text{m}$ and check whether Trouton's rule applies Use the given literature values to determine a theoretical value for the entropy of evaporation $\Delta_\text{vap} S_\text{m}$ .

The literature values are

\begin{aligned} \Delta_\text{vap} H_\text{m,acetone}^\text{lit.}&= 29{.}10\,\text{kJ}\,/\,\text{mol}\\ \Delta_\text{vap} H_\text{m,ethanol}^\text{lit.} &= 38{.}56\,\text{kJ}\,/\,\text{mol}\\ T_\text{b,acetone}^\text{lit.} &=329{.}20\,\text K\\ T_\text{b,ethanol}^\text{lit.} &=351{.}44\,\text K \end{aligned}