Question

A benzene-toluene solution with $x_{\text{benz }}=0.300$ has a normal boiling point of ${98.6}^{\circ }\mathrm{C}$. The vapor pressure of pure toluene at ${98.6}^{\circ }\mathrm{C}$ is $533\mathrm{mm}\mathrm{Hg}$. What must be the vapor pressure of pure benzene at ${98.6}^{\circ }\mathrm{C}?$ (Assume ideal solution behavior.)

Short answer

The vapor pressure of pure benzene at ${98.6}^{\circ }\mathrm{C}$ can be calculated using Raoult's law and the given information about the benzene-toluene solution. This involves four steps: understanding Raoult's law, applying Raoult's law to the benzene-toluene solution, calculating the mole fraction of toluene, and substituting the known variables into the equation to solve for $P_{\text{benzene}}^0$.

Step-by-step solution

Understand Raoult's Law

Raoults's law states that the partial pressure of each component of an ideal mixture of liquids is equal to the vapor pressure of the pure component multiplied by its mole fraction in the mixture. It can be represented as:$P_i=x_i\ast P_i^0$ where $P_i$ is the partial pressure of component i in the solution, $x_i$ is the mole fraction of component i in the solution and $P_i^0$ is the vapor pressure of pure component i.

Apply Raoult's Law for Benzene and Toluene

In the case of a benzene-toluene solution, the total pressure is equal to the sum of the partial pressures of benzene and toluene, i.e.$P_{\text{total}}=P_{\text{benzene}}+P_{\text{toluene}}$ which translates to $P_{\text{total}}=x_{\text{benzene}}\ast P_{\text{benzene}}^0+x_{\text{toluene}}\ast P_{\text{toluene}}^0$

Calculate the Mole Fraction of Toluene

The mole fraction of toluene can be calculated by subtracting the mole fraction of benzene from 1. In this case, $x_{\text{toluene}}=1-x_{\text{benzene}}=1-0.300=0.700$.

Substitute Known Variables and Solve for $P_{\text{benzene}}^0$

The next step is to substitute the known variables into the equation obtained in step 2 and solve for $P_{\text{benzene}}^0$:\n$P_{\text{total}}-x_{\text{toluene}}\ast P_{\text{toluene}}^0=x_{\text{benzene}}\ast P_{\text{benzene}}^0$\nAfter rearranging, the equation becomes\n$P_{\text{benzene}}^0=\frac{P_{\text{total}}-x_{\text{toluene}}\ast P_{\text{toluene}}^0}{x_{\text{benzene}}}$\nOn substituting the known values, $P_{\text{benzene}}^0$ can be calculated.

Key concepts

Ideal Solution

An ideal solution is a crucial concept in understanding mixtures and their behaviors. It refers to a mixture where the interactions between two different molecules are similar in strength to the interactions between molecules of the same kind. In simpler terms, the molecules mix without any strong or weak attractions compared to themselves. This results in predictable properties based on the components' properties alone.

In an ideal solution, the following assumptions are made:

• Molecules of different components obey simple arithmetic rules (e.g., they don't swell or contract significantly when mixed).
• Every component's contribution to the solution's properties is proportional to its concentration.
• There are no enthalpy changes – no heat is absorbed or released when components are mixed.

Such solutions follow Raoult's Law perfectly because the partial vapor pressures can be directly predicted by the mole fraction of each component, multiplied by its pure component vapor pressure.

Vapor Pressure

Vapor pressure is the pressure exerted by a vapor in equilibrium with its liquid (or solid) phase. It's a measure of a liquid's volatility – how readily it evaporates. Each pure substance has a specific vapor pressure at a given temperature, which increases with temperature.

For an ideal solution, Raoult's Law connects vapor pressures to mole fractions. It states that the partial vapor pressure of a component in the solution, say benzene or toluene, is given by:

• Multiplying the mole fraction of the component ( $x_i$) by the vapor pressure of the pure component ( $P_i^0$).
• The total vapor pressure over the solution is the sum of the partial pressures of all components.

Understanding this law helps predict how mixing impacts the vapor pressure of solutions, which is critical in refining, distillation, and chemical synthesis.

Mole Fraction

The mole fraction is a dimensionless quantity used in chemistry to describe the concentration of a component in a mixture. It is defined as the ratio of the number of moles of one component to the total number of moles of all components in the mixture.

It provides a straightforward way to express the composition of mixtures and solutions:

• For any component A, the mole fraction is given by: $x_A=\frac{n_A}{n_{total}}$, where $n_A$ represents the number of moles of component A and $n_{total}$ is the total number of moles of all components.
• Mole fractions are always less than or equal to 1, and the sum of the mole fractions in a mixture equals 1.

This means if you know the mole faction of one component, the other can be easily determined, as in our exercise where the mole fraction of benzene helps calculate the mole fraction of toluene.