Question

Styrene, used in the manufacture of polystyrene plastics, is made by the extraction of hydrogen atoms from ethylbenzene. The product obtained contains about \(38 \%\) styrene \(\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{CH}=\mathrm{CH}_{2}\right)\) and \(62 \%\) ethylbenzene \(\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{CH}_{2} \mathrm{CH}_{3}\right),\) by mass. The mixture is separated by fractional distillation at \(90^{\circ} \mathrm{C} .\) Determine the composition of the vapor in equilibrium with this \(38 \%-62 \%\) mixture at \(90^{\circ} \mathrm{C}\). The vapor pressure of ethylbenzene is \(182 \mathrm{mmHg}\) and that of styrene is \(134 \mathrm{mmHg}\).

Short answer

The pressure of the vapor above the mixture of styrene and ethylbenzene at \(90^{\circ} \mathrm{C}\) is \(163.6 \mathrm{mmHg}\).

Step-by-step solution

Calculating Mole Fractions

First, the exercise provides that the mixture is made up of 38% styrene and 62% ethylbenzene by mass. However, Raoult's law uses mole fractions hence we must convert these percentages into mole fractions. The molecular weight of styrene (C6H5CH=CH2) is \(104.15\, g/mol\) and that of ethylbenzene (C6H5CH2CH3) is \(106.17\, g/mol\). If we have 100 g of the mixture, then we will have \(38 g/104.15 g/mol = 0.365\, mol\) of styrene and \(62 g/106.17 g/mol = 0.584\, mol\) of ethylbenzene. The total amount of moles is \(0.365 mol + 0.584 mol = 0.949 mol\). Therefore, the mole fractions (X) of styrene and ethylbenzene are \(0.365 mol/0.949 mol = 0.385\) and \(0.584 mol/0.949 mol = 0.615\) respectively.

Applying Raoult's Law

Next, Raoult's Law can be used to determine the partial pressures of styrene and ethylbenzene in the mixture. Raoult's law states that the partial pressure of a component in a mixture is equal to the vapor pressure of the pure component multiplied by its mole fraction in the mixture. This gives us \(P_{Styrene} = X_{Styrene} * P^0_{Styrene} = 0.385 * 134 \,mmHg = 51.6\, mmHg\) and \(P_{Ethylbenzene} = X_{Ethylbenzene} * P^0_{Ethylbenzene} = 0.615 * 182 \,mmHg = 112\, mmHg\).

Calculating Total Pressure

The total pressure of the vapor in equilibrium with the liquid mixture is the sum of the partial pressures of the components. Using the values obtained in Step 2: \(P_{Total} = P_{Styrene} + P_{Ethylbenzene} = 51.6\, mmHg + 112\, mmHg = 163.6\, mmHg\).

Key concepts

Mole Fraction

The concept of mole fraction is crucial in understanding Raoult's Law, which is used to calculate the behavior of ideal mixtures. In any solution, the mole fraction is defined as the ratio of the moles of a particular component to the total number of moles of all components in the mixture.

For example, in a mixture consisting of styrene and ethylbenzene, if you have 0.365 moles of styrene and 0.584 moles of ethylbenzene, the total number of moles in the mixture is 0.949 moles. The mole fraction of styrene is thus calculated as 0.385, which is the division of its moles by the total moles (0.365/0.949). Similarly, the mole fraction for ethylbenzene is 0.615 (0.584/0.949).

This mole fraction helps in determining other important properties of the mixture, such as the partial pressure when coupled with Raoult's Law. Always remember to convert mass percentage into moles first, as Raoult's Law is based on mole fraction, not mass percentage. It is key in various applications such as chemical reactions, solutions, and phase equilibrium.

Partial Pressure

Partial pressure is a term used to describe the pressure exerted by each component of a gas mixture in a container. According to Raoult's Law, the partial pressure of a component in a liquid phase is proportional to the mole fraction of that component and its pure component's vapor pressure.

In the context of a styrene and ethylbenzene mixture, Raoult's Law is applied to find the partial pressures of each component. For instance, the partial pressure of styrene in the mixture will be calculated using its mole fraction (0.385) multiplied by its vapor pressure when pure, which is 134 mmHg. This results in a partial pressure of 51.6 mmHg. Similarly, the partial pressure for ethylbenzene is computed as 0.615 times 182 mmHg, yielding 112 mmHg.

The total pressure of the system in equilibrium is achieved by adding the partial pressures of styrene and ethylbenzene, giving a total of 163.6 mmHg in the provided mixture. Partial pressure is fundamental in predicting how different components in a mixture will interact, especially when assessing the composition of vapors over liquid mixtures.

Fractional Distillation

Fractional distillation is a common method used to separate mixtures based on different boiling points of the components in the mixture. It is particularly useful in chemical processes where a high degree of purity is necessary. In the case of a mixture of styrene and ethylbenzene, fractional distillation is used to isolate styrene from ethylbenzene, as their vapor pressures at a given temperature differ.

The principle behind fractional distillation relies on the fact that a component with a lower boiling point (or higher volatility) will vaporize more readily and thus will be enriched in the vapor phase. As this vapor is collected and condensed, it can be separated from the less volatile component, which remains in the liquid phase. Repeated distillations can increase the purity of the separated components.

In the scenario where styrene and ethylbenzene are being separated, the process exploits the partial pressures calculated using Raoult's Law. Because each component in the liquid mixture contributes to the total vapor pressure, adjusting conditions such as temperature can optimize the yields of the desired purified product through successive rounds of distillation.