Question

At \(25^{\circ} \mathrm{C},\) the vapor in equilibrium with a solution containing carbon disulfide and acetonitrile has a total pressure of 263 torr and is 85.5 mole percent carbon disulfide. What is the mole fraction of carbon disulfide in the solution? At \(25^{\circ} \mathrm{C}\) , the vapor pressure of carbon disulfide is 375 torr. Assume the solution and vapor exhibit ideal behavior.

Short answer

The mole fraction of carbon disulfide in the solution is approximately 0.6.

Step-by-step solution

Determine the partial pressure of carbon disulfide in the vapor

Given the total pressure (P_total) and the mole percentage of carbon disulfide in the vapor (85.5%), we can determine the partial pressure of carbon disulfide (P_CS2) using the following relationship: P_CS2 = P_total * (mole_percentage / 100) Where P_total = 263 torr and mole_percentage = 85.5%. Plugging in the values, we get: P_CS2 = 263 torr * (85.5 / 100) = 224.825 torr

Use Raoult's Law to determine the mole fraction of carbon disulfide in the solution

Raoult's Law states that the partial pressure of a component in the vapor phase is equal to the mole fraction of that component in the solution (x_CS2) times its vapor pressure (P*_CS2): P_CS2 = x_CS2 * P*_CS2 We know P_CS2 = 224.825 torr and P*_CS2 = 375 torr (given vapor pressure of carbon disulfide at 25°C). We can solve for x_CS2: x_CS2 = P_CS2 / P*_CS2 = 224.825 torr / 375 torr = 0.5994 Thus, the mole fraction of carbon disulfide in the solution is 0.5994, or approximately 0.6.

Key concepts

Partial Pressure

Partial pressure is an important concept in understanding gas mixtures and how different gases behave in a mix. When you have a mixture of gases, each gas contributes to the total pressure of the mixture. This contribution is the gas's partial pressure. It's as if each gas has its own private pressure it would exert if it were alone in the volume the mixture occupies.

To find the partial pressure of a gas in a mixture, you need to know the total pressure and the fraction of the mixture that is the target gas. The formula is simple:

• Partial Pressure of Gas = Total Pressure × Mole Fraction of the Gas in the Mixture

In the original exercise, the partial pressure of carbon disulfide was calculated by using the total pressure of the mixture and the mole percentage of carbon disulfide. This percentage is converted into a fraction, and then multiplied by total pressure to find the exact partial pressure contribution of carbon disulfide.

Calculating partial pressure is important, particularly when using it in conjunction with other laws like Raoult's Law to further explore solution properties.

Vapor Pressure

Vapor pressure is another key concept that helps to explain the behavior of solutions and mixtures. It refers to the pressure exerted by a vapor in equilibrium with its liquid or solid phase. In simple terms, it's the 'push' a liquid's molecules create when they escape into gaseous form.

Every liquid has a unique vapor pressure at any given temperature, depending on its molecular properties like intermolecular forces. Higher temperatures generally increase vapor pressure because more molecules have enough energy to enter the gas phase.

Vapor pressure becomes especially important in the application of Raoult's Law, where you'll multiply the mole fraction of a substance in a solution by its vapor pressure to determine the partial pressure contribution of that substance in the vapor phase. In the provided exercise, you used the given vapor pressure of carbon disulfide to calculate its mole fraction in the solution.

Understanding vapor pressure helps you predict how a substance will behave when it's part of a mixture, making it crucial for solving problems in chemistry that use Raoult's Law.

Mole Fraction

When dealing with solutions and mixtures, the mole fraction is a way to express the concentration of a component. It simply tells you the ratio of the moles of one component to the total moles of the mixture. Unlike percentages, which can be less precise, mole fractions are dimensionsless and give a more accurate idea of composition.

The mole fraction is considered very useful in calculations because it simplifies the math involved. For instance, in Raoult's Law, the mole fraction becomes part of the formula to determine the partial pressure of a component in a solution's vapor.

In the given exercise, finding the mole fraction of carbon disulfide involved using its partial pressure and the known total pressure. This shows you how mole fractions can bridge the gap between purely composition-focused measures and those that involve interactions like pressures.

Knowing how to calculate and apply mole fractions is essential, particularly when you need to extend beyond pure concentration and understand contributions to properties like vapor pressure.