Question

A solution is prepared by mixing 0.0300 mole of \(\mathrm{CH}_{2} \mathrm{Cl}_{2}\) and 0.0500 mole of \(\mathrm{CH}_{2} \mathrm{Br}_{2}\) at \(25^{\circ} \mathrm{C}\). Assuming the solution is ideal, calculate the composition of the vapor (in terms of molefractions) at \(25^{\circ} \mathrm{C} .\) At \(25^{\circ} \mathrm{C},\) the vapor pressures of pure \(\mathrm{CH}_{2} \mathrm{Cl}_{2}\) and pure \(\mathrm{CH}_{2} \mathrm{Br}_{2}\) are 133 and 11.4 torr, respectively.

Short answer

Using Raoult's Law and the given information, we first calculate the mole fractions of CH₂Cl₂ and CH₂Br₂ in the solution as \(X_{CH_2Cl_2} = 0.375\) and \(X_{CH_2Br_2} = 0.625\). Then, we calculate the partial pressures of each component, \(P_{CH_2Cl_2} = 49.875\, torr\) and \(P_{CH_2Br_2} = 7.125\, torr\). Finally, we find the mole fractions of each component in the vapor, \(Y_{CH_2Cl_2} = 0.875\) and \(Y_{CH_2Br_2} = 0.125\).

Step-by-step solution

Calculate the mole fraction of each component in the solution

First, we need to determine the mole fraction of each component in the solution. The mole fraction (X) is the ratio of the moles of a component (n) to the total moles of the solution (n_total). \[X_{CH_2Cl_2} = \frac{n_{CH_2Cl_2}}{n_{CH_2Cl_2} + n_{CH_2Br_2}}\] \[X_{CH_2Br_2} = \frac{n_{CH_2Br_2}}{n_{CH_2Cl_2} + n_{CH_2Br_2}}\] We are given the moles of each component, so we can calculate the mole fractions: \[n_{CH_2Cl_2} = 0.0300\, moles\] \[n_{CH_2Br_2} = 0.0500\, moles\] Calculate the mole fractions: \[X_{CH_2Cl_2} = \frac{0.0300}{0.0300 + 0.0500}\] \[X_{CH_2Br_2} = \frac{0.0500}{0.0300 + 0.0500}\]

Use Raoult's Law to calculate the partial pressures of each component

Raoult's Law states that the partial pressure (P) of a component in an ideal solution is equal to the product of its mole fraction (X) and the vapor pressure (P_vapor) of the pure substance: \[P_{CH_2Cl_2} = X_{CH_2Cl_2}\cdot P^{vapor}_{CH_2Cl_2}\] \[P_{CH_2Br_2} = X_{CH_2Br_2}\cdot P^{vapor}_{CH_2Br_2}\] Plug in the values for mole fractions and the vapor pressures of the pure substances: \[P^{vapor}_{CH_2Cl_2} = 133\, torr\] \[P^{vapor}_{CH_2Br_2} = 11.4\, torr\] \[P_{CH_2Cl_2} = X_{CH_2Cl_2}\cdot 133\, torr\] \[P_{CH_2Br_2} = X_{CH_2Br_2}\cdot 11.4\, torr\]

Calculate the mole fraction of each component in the vapor

Now that we have the partial pressures, we can calculate the mole fraction of each component in the vapor. To do this, we will use the formula: \[Y_{CH_2Cl_2} = \frac{P_{CH_2Cl_2}}{P_{CH_2Cl_2} + P_{CH_2Br_2}}\] \[Y_{CH_2Br_2} = \frac{P_{CH_2Br_2}}{P_{CH_2Cl_2} + P_{CH_2Br_2}}\] Plug in the partial pressure values calculated in Step 2: \[Y_{CH_2Cl_2} = \frac{P_{CH_2Cl_2}}{P_{CH_2Cl_2} + P_{CH_2Br_2}}\] \[Y_{CH_2Br_2} = \frac{P_{CH_2Br_2}}{P_{CH_2Cl_2} + P_{CH_2Br_2}}\]

Calculate the mole fractions of each component in the vapor

Using the mole fraction formulas and the partial pressure values, we can now calculate the mole fractions for each component in the vapor: \[Y_{CH_2Cl_2} = \frac{P_{CH_2Cl_2}}{P_{CH_2Cl_2} + P_{CH_2Br_2}}\] \[Y_{CH_2Br_2} = \frac{P_{CH_2Br_2}}{P_{CH_2Cl_2} + P_{CH_2Br_2}}\] These values represent the composition of the vapor at 25°C in terms of mole fractions.

Key concepts

Raoult's Law

Raoult's Law is a fundamental principle in physical chemistry that describes the relationship between vapor pressure and the composition of a liquid solution. It states that the partial pressure of each volatile component in a solution is directly proportional to its mole fraction in the mixture. This law applies to ideal solutions, where the interactions between different molecules are similar to the interactions between molecules of the same substance. In simple terms, according to Raoult's Law, each component of a solution contributes to the total vapor pressure based on its relative amount in the solution. Understanding this law is essential for studying boiling points, preparing mixtures, and analyzing distillation processes.

For instance, in the given exercise, you would use Raoult's Law to determine how much each component ((CH_2Cl_2) and (CH_2Br_2)) contributes to the overall vapor pressure of the solution. By knowing the pure vapor pressures and the mole fractions of each compound, you can calculate the partial pressures and, subsequently, the composition of the vapor above the solution.

Vapor Pressure

Vapor pressure is the pressure exerted by a vapor in equilibrium with its liquid or solid phase at a given temperature in a closed system. It is an intrinsic property of a substance, reflecting how volatile a substance is; the higher the vapor pressure, the more readily the substance will evaporate. When considering solutions, the vapor pressure above the solution is key to understanding the solution's behavior upon changes such as heating or mixing.

In the context of the exercise, the vapor pressures of pure (CH_2Cl_2) and (CH_2Br_2) are crucial data points. These pressures help us predict the behavior of the solution when temperature is held constant. When mixed, each substance's contribution to the total vapor pressure can be determined by applying Raoult's Law.

Partial Pressure

Partial pressure is the pressure that would be exerted by one of the gases in a mixture of gases if it alone occupied the entire volume of the original mixture at the same temperature. It's like imagining that all other gases are removed without changing the temperature or volume and seeing how much pressure the remaining gas would exert. This concept is useful in various scientific fields, from respiratory physiology to chemical engineering.

In our exercise, after calculating the mole fractions using the moles provided, we use them to find the partial pressures of (CH_2Cl_2) and (CH_2Br_2) in the solution. This step is done by multiplying the mole fraction of each compound by its respective vapor pressure.

Ideal Solution Behavior

An ideal solution is one where the intermolecular forces between the solute and solvent are equal to the forces between the molecules of the pure substances. Here, the mix of solvents doesn't change physical properties such as enthalpy; it behaves as though each component is alone in the solution. Ideal solutions follow Raoult's Law perfectly because the interactions among different particles don't differ from the pure substances. They are mostly theoretical because most real solutions have at least some deviation from this idealized behavior.

For the problem we're discussing, assuming ideal behavior simplifies the calculations because it implies that the mole fractions and vapor pressures will directly give us the partial pressures and vapor composition without needing to account for interaction differences between molecules of (CH_2Cl_2) and (CH_2Br_2).