Problem 97
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.
Step-by-Step Solution
Verified Answer
The mole fraction of carbon disulfide in the solution is approximately 0.60.
1Step 1: Calculate partial pressure of carbon disulfide in the vapor
Since we know the mole percentage of carbon disulfide in the vapor, we can calculate its partial pressure. To do this, multiply the total pressure of the vapor by the mole percentage of carbon disulfide.
Partial pressure of carbon disulfide (P_CS2) = Total pressure × Mole percentage of carbon disulfide
\(P_{CS_2} = 263 \, \text{torr} \times 0.855\)
2Step 2: Calculate the mole fraction of carbon disulfide in the vapor
To determine the mole fraction of carbon disulfide in the vapor (y_CS2), we can use its partial pressure (from step 1) and the vapor pressure of pure carbon disulfide.
Using Raoult's Law,
\(P_{CS_2} = x_{CS_2} \times P^*_ {CS_2}\)
where P_CS2 is the partial pressure of carbon disulfide, x_CS2 is the mole fraction of carbon disulfide in the solution, and \(P^*_{CS_2}\) is the vapor pressure of pure carbon disulfide.
Rearrange the equation to find the mole fraction of carbon disulfide in the solution:
\(x_{CS_2} = \frac{P_{CS_2}}{P^*_{CS_2}}\)
3Step 3: Calculate the mole fraction of carbon disulfide in the solution
Now, we can calculate the mole fraction of carbon disulfide in the solution (x_CS2) using the equation from step 2.
\(x_{CS_2} = \frac{P_{CS_2}}{P^*_{CS_2}}\)
Calculate \(P_{CS_2}\) from step 1:
\(P_{CS_2} = 263 \, \text{torr} \times 0.855 = 224.945 \, \text{torr}\)
Now, use \(P^*_{CS_2} = 375 \, \text{torr}\) (given) and substitute the values into the equation:
\(x_{CS_2} = \frac{224.945 \, \text{torr}}{375 \, \text{torr}}\)
4Step 4: Calculate the final result
Calculate the mole fraction of carbon disulfide in the solution:
\(x_{CS_2} = \frac{224.945 \, \text{torr}}{375 \, \text{torr}} = 0.59986\)
The mole fraction of carbon disulfide in the solution is approximately 0.60.
Key Concepts
Raoult's LawPartial PressureVapor PressureIdeal Solution Behavior
Raoult's Law
Raoult's Law is a principle of physical chemistry that is essential for understanding the behavior of solutions and their associated vapor pressures. Let's clarify it with simple language.
Consider a liquid mixture made up of more than one component, like a soft drink which contains various flavors mixed with water. Raoult's Law states that the vapor pressure of each component of an ideal mixture is directly proportional to the mole fraction of the component in the mixture (just like how the taste of our soft drink depends on the proportion of each flavor). Mathematically, it's written as:
\[ P_i = X_i \times P^{\ast}_i \]
where \( P_i \) is the partial vapor pressure of component \( i \), \( X_i \) is the mole fraction of component \( i \) in the liquid phase, and \( P^{\ast}_i \) is the vapor pressure of the pure component \( i \). This straightforward relationship allows us to determine the composition of a solution's vapor based on the known properties of its individual components. In the textbook exercise, Raoult's Law is used to relate the partial pressure of carbon disulfide in the vapor to its mole fraction in the solution.
Consider a liquid mixture made up of more than one component, like a soft drink which contains various flavors mixed with water. Raoult's Law states that the vapor pressure of each component of an ideal mixture is directly proportional to the mole fraction of the component in the mixture (just like how the taste of our soft drink depends on the proportion of each flavor). Mathematically, it's written as:
\[ P_i = X_i \times P^{\ast}_i \]
where \( P_i \) is the partial vapor pressure of component \( i \), \( X_i \) is the mole fraction of component \( i \) in the liquid phase, and \( P^{\ast}_i \) is the vapor pressure of the pure component \( i \). This straightforward relationship allows us to determine the composition of a solution's vapor based on the known properties of its individual components. In the textbook exercise, Raoult's Law is used to relate the partial pressure of carbon disulfide in the vapor to its mole fraction in the solution.
Partial Pressure
The concept of partial pressure is akin to a group project where each member contributes a 'part' of the total effort. In a mixture of gases, each gas exerts its own pressure independently as if the other gases were not present. This independent pressure is known as the partial pressure.
When multiple gases coexist in a container, the total pressure is the sum of the individual pressures that each gas would exert if it alone occupied the entire volume. This is encapsulated by Dalton's Law:
\[ P_{\text{total}} = P_1 + P_2 + P_3 + ... \]
For instance, if you're inflating a balloon with a mixture of helium and oxygen, the total pressure inside the balloon is the sum of the partial pressure of helium and the partial pressure of oxygen. In our textbook example, we're focused on the pressure contributed by carbon disulfide when mixed with acetonitrile, recognizing that it is a crucial part of the total pressure we read on a barometer or other measuring devices.
When multiple gases coexist in a container, the total pressure is the sum of the individual pressures that each gas would exert if it alone occupied the entire volume. This is encapsulated by Dalton's Law:
\[ P_{\text{total}} = P_1 + P_2 + P_3 + ... \]
For instance, if you're inflating a balloon with a mixture of helium and oxygen, the total pressure inside the balloon is the sum of the partial pressure of helium and the partial pressure of oxygen. In our textbook example, we're focused on the pressure contributed by carbon disulfide when mixed with acetonitrile, recognizing that it is a crucial part of the total pressure we read on a barometer or other measuring devices.
Vapor Pressure
Vapor pressure is a manifestation of a liquid's tendency to convert into vapor, which can be likened to a desire of water to escape from a boiling pot into the air as steam. More scientifically, vapor pressure is the pressure exerted by a vapor when it is in equilibrium with its liquid or solid form, meaning the rate at which molecules evaporate from the liquid equals the rate at which they condense back into it.
The vapor pressure of a substance depends on its temperature. As the temperature increases, so does the vapor pressure, because more molecules have enough energy to escape into the vapor phase. Every substance has its own unique vapor pressure curve, which graphically represents this relationship. For pure substances, this is a constant value at a given temperature, like the 375 torr for pure carbon disulfide at 25°C described in the exercise. It's vital to understanding how a substance will behave under different conditions and plays a pivotal role in calculations involving Raoult's Law.
The vapor pressure of a substance depends on its temperature. As the temperature increases, so does the vapor pressure, because more molecules have enough energy to escape into the vapor phase. Every substance has its own unique vapor pressure curve, which graphically represents this relationship. For pure substances, this is a constant value at a given temperature, like the 375 torr for pure carbon disulfide at 25°C described in the exercise. It's vital to understanding how a substance will behave under different conditions and plays a pivotal role in calculations involving Raoult's Law.
Ideal Solution Behavior
An ideal solution is like the perfect couple, where each component's presence doesn't alter the other's behavior; they're in complete harmony. More technically, an ideal solution is one where the interactions between molecules of different components are the same as the interactions between molecules of the same component. This means that mixing them doesn't result in any heat being absorbed or released, and the volume doesn't change.
In such a solution, Raoult's Law can be applied without any corrections. Ideal solutions follow the law perfectly because their vapor pressures are directly proportional to the mole fractions of the individual components. In our exercise, the assumption of ideal behavior simplifies calculations because we can directly relate the total vapor pressure, the vapor pressures of the individual components, and the mole fractions without adjusting for any complex interactions. Understanding ideal solutions and their properties is fundamental for grasping more complicated non-ideal mixtures where interactions can dramatically alter the expected outcomes.
In such a solution, Raoult's Law can be applied without any corrections. Ideal solutions follow the law perfectly because their vapor pressures are directly proportional to the mole fractions of the individual components. In our exercise, the assumption of ideal behavior simplifies calculations because we can directly relate the total vapor pressure, the vapor pressures of the individual components, and the mole fractions without adjusting for any complex interactions. Understanding ideal solutions and their properties is fundamental for grasping more complicated non-ideal mixtures where interactions can dramatically alter the expected outcomes.
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