Question

The vapor pressure of pure \(\mathrm{CCl}_{4}\) at \(65^{\circ} \mathrm{C}\) is \(504 \mathrm{~mm} \mathrm{Hg}\). How many grams of naphthalene \(\left(\mathrm{C}_{10} \mathrm{H}_{8}\right)\) must be added to \(25.00 \mathrm{~g}\) of \(\mathrm{CCl}_{4}\) so that the vapor pressure of \(\mathrm{CCl}_{4}\) over the solution is \(483 \mathrm{~mm}\) Hg? Assume that the vapor pressure of naphthalene at \(65^{\circ} \mathrm{C}\) is negligible.

Short answer

(Vapor pressure of pure 𝖢𝖢𝗅4 is 504 mm Hg) Answer: Approximately 0.913 grams of naphthalene should be added to the 25.00 grams of 𝖢𝖢𝗅4 to achieve the desired vapor pressure.

Step-by-step solution

Write down the equation for Raoult's Law

Raoult's Law is given by the equation: \(P_{A} = X_{A} \times P_{A}^{*}\), where \(P_{A}\) is the vapor pressure of component A in the solution, \(X_{A}\) is the mole fraction of component A, and \(P_{A}^{*}\) is the vapor pressure of pure component A.

Calculate the initial mole fraction of 𝖢𝖢𝗅4

Initially, there are no other components in the solution, so the mole fraction of 𝖢𝖢𝗅4 is 1.

Calculate the final mole fraction of 𝖢𝖢𝗅4

We are given that the vapor pressure of 𝖢𝖢𝗅4 over the solution is supposed to be 483 mm Hg. Using Raoult's Law, we can substitute the values and calculate the final mole fraction of 𝖢𝖢𝗅4: \(483 = X_{CCl4} \times 504\) To find \(X_{CCl4}\), we divide both sides of the equation by 504: \(X_{CCl4} = \frac{483}{504} = 0.9583\)

Determine the mass of naphthalene (𝖢10𝖧8) required

Now we need to find the mass of naphthalene that needs to be added to achieve this mole fraction. To do this, we need to find the ratio between moles of naphthalene and moles of 𝖢𝖢𝗅4. Since the mole fraction of 𝖢𝖢𝗅4 is 0.9583, the mole fraction of naphthalene is 1 - 0.9583 = 0.0417 Now we can use the ratio to find the amount of naphthalene: \(\frac{moles\ of\ C10H8}{moles\ of\ CCl4} = \frac{0.0417}{0.9583}\) The moles of 𝖢𝖢𝗅4 can be calculated using the mass given and the molar mass of 𝖢𝖢𝗅4 (153.8 g/mol): \(moles\ of\ CCl4 = \frac{25.00g}{153.8g/mol} = 0.1625\ mol\) Using the ratio of moles: \(moles\ of\ C10H8 = 0.1625\ mol \times \frac{0.0417}{0.9583} = 0.00713\ mol\) Now we can find the mass of naphthalene (molar mass of 𝖢10𝖧8 = 128.2 g/mol): \(mass\ of\ C10H8 = 0.00713\ mol \times 128.2g/mol = 0.913\ g\) To achieve the desired vapor pressure, approximately 0.913 grams of naphthalene must be added to the 25.00 grams of 𝖢𝖢𝗅4.

Key concepts

Understanding Vapor Pressure

Vapor pressure is a critical property of liquids that dictates how a substance will behave when mixed with others in a solution. It refers to the pressure exerted by a vapor in equilibrium with its liquid or solid form at a given temperature. In other words, it's the force the gaseous form of a substance exerts when some of the particles escape from the liquid or solid phase into the air above it. The equilibrium is reached when the rate of evaporation equals the rate of condensation.

The vapor pressure of a pure substance is constant at a given temperature. However, when a non-volatile solute (a solute that does not readily evaporate and form a gas) is added to the solvent, the vapor pressure of the resulting solution is lower than that of the pure solvent. This phenomenon is because the solute particles occupy space at the surface of the liquid and, therefore, fewer solvent molecules can escape into the gas phase. The presence of a non-volatile solute essentially decreases the number of solvent molecules available to evaporate, which in turn lowers the vapor pressure.

Determining Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a mixture. It's defined as the ratio of the number of moles of a particular substance to the total number of moles of all substances present in the mixture. Mathematically, for a component A, it is expressed as:
\[ X_A = \frac{n_A}{n_{total}} \]
where \(X_A\) is the mole fraction of substance A, \(n_A\) is the number of moles of A, and \(n_{total}\) is the total number of moles of all components.

The mole fraction is a unitless quantity and it always lies between 0 and 1. When dealing with solutions, the mole fraction can give you an insight into the composition of the solution. For example, if a solution has a mole fraction of a component close to 1, that component is the primary substance in the mixture.

Solution Concentration: Core Element in Raoult's Law

Solution concentration plays a fundamental role in determining the properties of solutions, such as vapor pressure as described by Raoult's Law. Raoult's Law connects the vapor pressure of the solvent in a solution to the mole fraction of the solvent in the solution. It is applicable to ideal solutions, where the interactions between different particles are similar to those when they are separate.

According to Raoult's Law, the vapor pressure of the solvent above a solution \(P_A\) is equal to the vapor pressure of the pure solvent \(P_A^*\) multiplied by the mole fraction of the solvent \(X_A\) in the mixture:
\[ P_{A} = X_{A} \times P_{A}^{*} \]
When calculating the concentration of a solution, it's not just about the amount of solute and solvent present, but also about their molecular weight which affects the number of moles and consequently the mole fraction of each component. This is why, in our exercise example, understanding the relationship between grams, moles, and mole fraction was essential to solve the problem, ensuring the correct amount of naphthalene was added to achieve the desired vapor pressure.