Partial molar volume - Physical Chemistry Lab Course

Theory

The aim is to determine the partial molar volume of water and sodium hydroxide within a binary mixture of the two components. The partial molar volume is then compared to the molar volume of the pure substances.

Molar volume

The molar volume of a pure substance $V_\text{m}^\text{pure}$ is the ratio of volume $V$ and the amount of substance $n$ :

V_\text{m}^\text{pure} = \frac{V}{n}

Generally, the molar volume $V_\text{m}$ depends on pressure $p$ and temperature $T$ , since its volume $V$ is pressure and temperature dependent. The molar volume of the ideal gas $V_\text{m}^\text{ideal}$ is

V_\text{m}^\text{ideal} = \frac{RT}{p},

since interaction between the gas molecules are neglected. The universal gas constant is $R=8{.}314\,\text J\,/\,(\text{mol}\,\text K)$ . If the interactions between the particles of a pure substances cannot be neglected, the molar volume (equation ) deviates from the molar volume of an ideal gas (equation ). For liquids and solids the deviation is particularly pronounced.

Definition of the partial molar volume

Next we consider a homogeneous mixture of $N$ different pure substances. The volume of the mixture depends on pressure and temperature. Furthermore, experimental data shows that the volume of the mixture additionally depends on the composition of the mixture.

V = V(p, T, n_1, n_2, ..., n_N)\,.

The total differential of the volume of the mixture is:

\mathrm d V = \left(\frac{\partial V}{\partial p}\right)_{T,n_i}\cdot \mathrm d p + \left(\frac{\partial V}{\partial T}\right)_{p,n_i}\cdot \mathrm d T + \sum_{i=1}^N \left(\frac{\partial V}{\partial n_i}\right)_{p,T,n_{i\neq j}}\cdot \mathrm d n_i

Here, we denote the change in the mixture volume upon a change of the amount of substance of component $i$ as partial molar volume $V_{\text{m},i}$ .

\left(\frac{\partial V}{\partial n_i}\right)_{p,T,n_{i\neq j}} := V_{\text{m},i}

The subscript $p,T,n_{i\neq j}$ means that pressure, temperature and amount of substance of each other component is held constant. If the interaction between the particles of the mixture would be the same as within the pure substance, the partial molar volume would not depend on the composition and would thus equal the molar volume of the pure substance. In this case, the volume of the mixture would be the sum of the molar volume of the pure substances. If, however, there are different interactions between the particles of the mixture, the partial molar volumes $V_{\text{m},i}$ are different from the molar volumes $V_{\text{m},i}^{\text{pure}}$ of the pure substances:

V_{\text{m},i} = V_{\text{m},i}(p, T, n_1, n_2, ..., n_N)\neq V_{\text{m},i}^{\text{pure}}

The partial molar volumes are a function of pressure, temperature and composition of the mixture.

Binary mixtures

In this lab course a homogeneous mixture of water and sodium hydroxide is investigated. In this two-component mixture ( $N=2$ ) a change of the amount of substance of a single component ( $\mathrm d n_1\neq 0$ ) under constant pressure ( $\mathrm d p=0$ ) and constant temperature ( $\mathrm d T=0$ ) while keeping the amount of substance of the other component constant ( $\mathrm d n_2=0$ ) allows to determine the partial molar volume of the first component using equation .

V_{\text{m},1}=\left(\frac{\partial V}{\partial n_1}\right)_{p,T,n_2}\approx \frac{V_\text{diff}}{n_\text{1,diff}}

For this it is necessary to measure the change in volume of the mixture $V_\text{diff}$ upon addition of an amount of substance of the first component $n_\text{1,diff}$ . Since the volume $V$ of the binary mixture is

V=V_{\text{m},1}\,n_1 + V_{\text{m},2}\,n_2,

we can also determine the partial molar volume of the second component if we know each component's total amount of substance $n_1$ and $n_2$ :

V_{\text{m},2}=\frac{V-V_{\text{m},1}\,n_1}{n_2}\approx\frac{V}{n_2}- \frac{V_\text{diff}}{n_\text{1,diff}}\frac{n_1}{n_2}

By comparing the experimentally determined partial molar volumes from equations and with the molar volumes of the pure substances from equation we can determine whether, for example, the interaction between two water molecules is different compared to the interaction between water and sodium and hydroxide ions, respectively.

Experimental setup

Volume measurement apparatus

The apparatus for mixing the two components and measuring the volume of the mixture is a vessel with a burette. A tap on the bottom of the vessel allows to fill the vessel with water from a reservoir. Using a funnel, the solid sodium hydroxide can be added to the water. The volume of the mixture is determined using the burette. The vessel is surrounded by a tempering jacket and additionally a heat exchange spiral is inside the vessel. This allows to keep the mixture at a constant Celsius temperature of 25 ℃ using a thermostat. The temperature can be checked using a thermometer, which is placed in a small glass inlet filled with water (such that the probe is not in contact with the sodium hydroxide solution). Below the vessel, a magnetic stirrer allows to speed up the dissolution of sodium hydroxide.

Degassing apparatus

At large concentration of sodium hydroxide the solubility of air in the mixture is less than in pure water. Using non-degassed (i.e. normal) water gas bubbles would form in the course of the experiment that influence the volume measurement. For this, deionized water is degassed with a diaphragm pump under stirring.

Scale

Solid sodium hydroxide is hygroscopic and is therefore portioned in a screw cap bottle. Determine the mass of the filled bottle. After adding the solid to the mixture the actually used mass is determined by weighing the empty bottle again and taking the difference. To do this, the empty container including the top and funnel is filled with the corresponding amount of sodium hydroxide without taring and the mass is noted as $m_\text{voll}$ . After the container has been emptied, the empty container, including the top and funnel, is weighed again.

Instructions

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Lab course instructions

Preparation

Turn on the thermostat and set the Celsius temperature to 25 ℃. Make sure the cooling circulation is active.
Fill the vessel with demineralised water until the burette reaches a level of 3-4 mL. Open the three-way valve in the direction of the burette and Woulf's bottle and closed to the outside air at the same time. Switch on the magnetic stirrer and degassing takes place with the vessel closed after the pump is switched on. Degassing takes place for 10-15 minutes.
Then switch off the pump and carefully turn the three-way valve to allow the vessel access to the outside air so that the vessel is ventilated.
After the Celsius temperature has stabilised, write down the Celsius temperature, the volume of the burette and the ambient pressure.

Measuring the volume as function of added sodium hydroxide mass

Weigh a portion of approximately 10 g to 15 g of sodium hydroxide in the screw-top bottle. Work quickly, as sodium hydroxide is hygroscopic. Weigh the filled bottle, transfer the sodium hydroxide slowly and carefully via the burette into the vessel and weigh the empty bottle. The mass of the added sodium hydroxide is later determined by taking the mass difference.
Wait until the mixture is homogeneous and the temperature is constant. Note the Celsius temperature, the pressure and the burette volume. Make sure that the thermometer is immersed in the contact liquid (water).
If, after some time, there are still some pieces of sodium hydroxide left (not dissolved), note down the number of sodium hydroxide pieces and determine later the mass of a single sodium hydroxide piece to consider the number of undissolved pieces later in the analysis.
Repeat the above steps for a total of 8 additions of sodium hydroxide.

Clean up

Turn off the thermostat and remove the mixture from the vessel.

Clean the vessel multiple times with deionized water. Take care that no sodium hydroxide pieces remain at the scale.

Clean up all equipment that has been in contact with sodium hydroxide.

Analysis

Molar volume of the pure substances

Use equation and the following values for the molar mass $M$ and density $\rho$ to compute the molar volume of solid sodium hydroxide (index 1) and water (index 2) at $\vartheta=25°\text C$ and $p=101325\,\text{Pa}$ . Compare the computed molar volumes with the molar volume of the ideal gas (equation ).

\begin{aligned} M_{\text{1}} &= 39{.}997\,\text g / \text{mol}, &\rho_{\text{1}} &= 2{.}13\,\text g / \text{cm}^3\\ M_{\text{2}} &= 18{.}015\,\text g / \text{mol}, &\rho_{\text{2}} &= 0{.}997\,\text g / \text{cm}^3 \end{aligned}

Assume that the given values for the molar mass and the density are with error/uncertainty.

Calculation of the mixture volume

Determine the volume of the mixture by adding the measured burette volume to the volume of the vessel. At a burette volume of $V_{\text{burette}} = 0\,\text{mL}$ the vessel volume is $V=V_\text{vessel} = (994\pm0{.}05)\,\text{mL}$ . Compute the total amount of substance of sodium hydroxide $n_1$ after each addition of sodium hydroxide. Plot the mixture volume $V$ (ordinate) against the amount of substance of sodium hydroxide $n_1$ (abscissa) and discuss the data. Compare with the volume of solid sodium hydroxide added to the mixture.

Determination of the partial molar volume

Determine the partial molar volume of sodium hydroxide $V_{\text{m},1}$ using equation and the partial molar volume of water $V_{\text{m},2}$ with equation . Calculate from the initial amount of substance of water $n_2$ and the respective amount of substance of sodium hydroxide $n_1$ in the mixture the mole fraction of sodium hydroxide $x_1$ . Plot the calculated partial molar volume (ordinate) against the mole fraction of sodium hydroxide (abscissa) in a single diagram. Discuss your data and compare with the molar volumes of the pure substances.