Question

\(\beta \)-Pinene \(\left(C_{10} H_{16}\right)\) and \(\alpha\) -terpineol \(\left(C_{10} H_{18} O\right)\) are used in cosmetics to provide a "fresh pine" scent. At \(367 \mathrm{~K},\) the pure substances have vapor pressures of 100.3 torr and 9.8 torr, respectively. What is the composition of the vapor (in terms of mole fractions) above a solution containing equal masses of these compounds at \(367 \mathrm{~K} ?\) (Assume ideal behavior.)

Short answer

The vapor composition is approximately 92.1% \(\beta\)-Pinene and 7.9% \(\alpha\)-terpineol.

Step-by-step solution

- Calculate the number of moles

Determine the molar masses of \(\beta\)-Pinene \(C_{10}H_{16}\) and \(\beta\)-terpineol \left(C_{10} H_{18} O\right)\. The molar mass of \(C_{10}H_{16}\) is \( 10 \times 12 + 16 \times 1 = 136 \text{g/mol}\). The molar mass of \(C_{10} H_{18} O\) is \(10 \times 12 + 18 \times 1 + 16 = 154 \text{g/mol}\)\. Given equal masses of both compounds, let's assume we have 1 gram of each.

- Calculate moles of each compound

Using the molar masses calculated, the moles of \(C_{10} H_{16}\) are \( \frac{1}{136} \approx 0.00735 \text{mol}\). For \(C_{10} H_{18} O\), the moles are \( \frac{1}{154} \approx 0.00649 \text{mol}\)\.

- Determine the mole fraction in the liquid phase

Calculate the total moles: \( 0.00735 + 0.00649 = 0.01384 \text{mol}\). The mole fraction of \(C_{10} H_{16}\) is \( \frac{0.00735}{0.01384} \approx 0.531 \) and the mole fraction of \(C_{10} H_{18} O\) is \( \frac{0.00649}{0.01384} \approx 0.469 \)\.

- Apply Raoult's Law to find partial pressures

Using Raoult's Law: \( P_A = P_A^0 \times x_A \) and \( P_B = P_B^0 \times x_B \). For \(C_{10} H_{16}\), partial pressure \( P_A = 100.3 \text{torr} \times 0.531 \approx 53.28 \text{torr}\)\. For \(C_{10} H_{18} O\), partial pressure \( P_B = 9.8 \text{torr} \times 0.469 \approx 4.60 \text{torr}\)\.

- Calculate the total vapor pressure

Add the partial pressures to get the total vapor pressure: \( P_{total} = 53.28 + 4.60 = 57.88 \text{torr}\)\.

- Determine vapor-phase mole fractions

Calculate the mole fraction of \(C_{10} H_{16}\) in vapor: \( y_A = \frac {P_A}{P_{total}} = \frac{53.28}{57.88} \approx 0.921 \). For \(C_{10} H_{18} O\), the mole fraction is \( y_B = \frac{P_B}{P_{total}} = \frac{4.60}{57.88} \approx 0.079 \).

Key concepts

Vapor Pressure

Vapor pressure is the pressure exerted by a vapor in equilibrium with its liquid phase at a given temperature. It is an essential property in understanding how substances evaporate and behave in a mixture. When a liquid's molecules escape into the gas phase, they exert a pressure known as vapor pressure.
The vapor pressure of a substance depends on its temperature and its inherent physical properties. Higher temperatures lead to higher vapor pressures as more molecules have enough energy to escape into the gas phase.
In our example, at 367 K, pure \( \beta \) -Pinene has a vapor pressure of 100.3 torr, and \( \alpha \) -Terpineol has a vapor pressure of 9.8 torr. These values are critical because they are used later in Raoult's Law to find the partial pressures in a mixture.

Ideal Solution

An ideal solution behaves in a way that the interactions between different chemical species are similar to the interactions between identical species. This means the mixing of compounds does not involve any energy change (enthalpy changes are negligible).
Ideal solutions follow Raoult's Law, which states that the partial vapor pressure of each component in the solution is directly proportional to its mole fraction in the mixture and its pure component vapor pressure.
For our example, we assume ideal behavior and use Raoult's Law. We consider the vapor pressures of pure \( \beta \) -Pinene and \( \alpha \) -Terpineol and their respective mole fractions to calculate the partial pressures of each component in the mixture. This assumption significantly simplifies the calculations, allowing us to use straightforward multiplication to find each compound's partial pressure.

Mole Fraction

Mole fraction is a way of expressing the concentration of a component in a mixture. It is defined as the ratio of the number of moles of a particular component to the total number of moles in the mixture.
Mathematically, for a component A in a mixture containing components A and B:
\[ x_A = \frac{{n_A}}{{n_A + n_B}} \]
where \( n_A \) and \( n_B \) are the moles of components A and B, respectively.
In our problem, given equal masses of \( \beta \) -Pinene and \( \alpha \) -Terpineol, we first calculate the moles of each using their molar masses. From these moles, we determine the mole fractions in the liquid phase:
\[ x_A = \frac{{0.00735}}{{0.00735 + 0.00649}} \approx 0.531 \]
\[ x_B = \frac{{0.00649}}{{0.00735 + 0.00649}} \approx 0.469 \]
These mole fractions are then used in Raoult's Law to find the partial pressures and, finally, the mole fractions of each component in the vapor phase above the solution.