Problem 64 - Chemistry The Central Science

Question

(a) What is an ideal solution? (b) The vapor pressure of pure water at \(60^{\circ} \mathrm{C}\) is 149 torr. The vapor pressure of water over a solution at \(60^{\circ} \mathrm{C}\) containing equal numbers of moles of water and ethylene glycol (a nonvolatile solute) is 67 torr. Is the solution ideal according to Raoult's law? Explain.

Step-by-Step Solution

Answer

An ideal solution is a mixture of two or more substances that follows Raoult's law at all concentrations. The given solution contains equal mole fractions of water and ethylene glycol. According to Raoult's law, the expected vapor pressure of water over the ideal solution would be 74.5 torr. However, the measured vapor pressure is 67 torr, which significantly deviates from the expected value. Therefore, the solution is not ideal according to Raoult's law.

Step 1: (1) Definition of an Ideal Solution

An ideal solution is a mixture of two or more substances that follows Raoult's law at all concentrations. Raoult's law states that the partial vapor pressure of each component in an ideal solution is proportional to the mole fraction of that component. In mathematical terms, for component A in an ideal solution: \[\ P_A = \chi_A P_A^* \] Here, \(P_A\) is the partial vapor pressure of component A, \(\chi_A\) is the mole fraction of A in the solution, and \(P_A^*\) is the vapor pressure of the pure component A.

Step 2: (2) Calculate the mole fraction

We are given that the solution contains equal numbers of moles of water and ethylene glycol. Therefore, the mole fractions of each component are: \[\chi_{H_2O} = \frac{1}{1 + 1} = 0.5\] \[\chi_{C_2H_6O_2} = \frac{1}{1 + 1} = 0.5\]

Step 3: (3) Calculate vapor pressure according to Raoult's law

Now, we can use Raoult's law to calculate the vapor pressure of water over an ideal solution: \[P_{H_2O} = \chi_{H_2O} * P_{H_2O}^*\] \[P_{H_2O} = 0.5 * 149\,\text{torr} = 74.5\,\text{torr}\]

Step 4: (4) Compare measured and calculated vapor pressures

The measured vapor pressure over the solution containing water and ethylene glycol is 67 torr, while the expected vapor pressure of water over an ideal solution according to Raoult's law is 74.5 torr. Since the measured vapor pressure is significantly lower than the expected vapor pressure, the solution is not ideal according to Raoult's law.

Step 5: (5) Conclusion

An ideal solution is one that follows Raoult's law. In the given problem, the solution containing water and ethylene glycol does not show vapor pressures consistent with Raoult's law, indicating that the solution is not ideal.

Key Concepts

Ideal Solution

An ideal solution is a special type of mixture that behaves perfectly according to a specific rule known as Raoult's law. This means that in such a solution, the chemical interactions between different types of molecules are the same as the interactions between molecules of the same type.
The key point to understand is that an ideal solution behaves as if each type of molecule simply "ignores" the fact that it is surrounded by different molecules. This allows the solution to have predictable properties.
One main feature of an ideal solution is its ability to follow Raoult's law across any concentration of the components involved.

• Raoult's law states: The partial vapor pressure of any component in an ideal solution is directly proportional to its mole fraction in the mixture.
• If everything behaves ideally, mixing components won't result in any heat being absorbed or released, and volume changes won't occur.

While the concept is straightforward, not all mixtures follow Raoult's law due to intermolecular attractions and other complex behaviors.

Vapor Pressure

Vapor pressure is an essential concept when exploring solutions. It is the pressure exerted by the vapor present above a liquid in a closed system at equilibrium.
This is especially important in solutions, where vapor pressure helps us understand how components evaporate and behave.

• In pure substances, vapor pressure depends on the temperature and the substance's inherent properties.
• For solutions, especially when evaluating ideal solutions, Raoult's law helps predict the vapor pressure when a solute is added to a solvent.

When dealing with an ideal solution, changes in vapor pressure can be accurately predicted using Raoult’s law.
It shows us how the presence of a solute affects the vapor pressure of the solvent, often leading to a decrease as seen in the example of water and ethylene glycol.
The measured vapor pressure deviating from the calculated value provides evidence of non-ideal behavior.

Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a solution. It shows how many moles of a specific substance are present relative to the total number of moles in the mixture.
This measurement is used to express the proportion of each component in a solution, and it is particularly useful in calculations involving vapor pressure and Raoult's law.

• The formula for mole fraction, \( \chi \), of a component A in a solution is given by: \( \chi_A = \frac{n_A}{n_A + n_B} \) where \( n_A \) and \( n_B \) are the moles of components A and B respectively.
• In the problem discussed, water and ethylene glycol have equal mole fractions, each being 0.5.

By simply knowing the mole fractions and the vapor pressures of pure components, one can apply Raoult's law to determine the expected vapor pressure of a solution, highlighting the importance of mole fraction in characterizing solution behavior.