Plot the vapour pressure data for a mixture of benzene (B) and acetic acid (A) given below and plot the vapour pressure/composition curve for the mixture at \(50^{\circ} \mathrm{C}\). Then confirm that Raoult's and Henry's laws are obeyed in the appropriate regions. Deduce the activities and activity coefficients of the components on the Raoult's law basis and then, taking \(\mathrm{B}\) as the solute, its activity and activity coefficient on a Henry's law basis. Finally, evaluate the excess Gibbs energy of the mixture over the composition range spanned by the data. $$ \begin{array}{lllllll} x_{\mathrm{A}} & 0.0160 & 0.0439 & 0.0835 & 0.1138 & 0.1714 & \\ P_{\mathrm{A}} / \mathrm{kPa} & 0.484 & 0.967 & 1.535 & 1.89 & 2.45 & \\ P_{\mathrm{J}} / \mathrm{kPa} & 35.05 & 34.29 & 33.28 & 32.64 & 30.90 & \\ x_{\mathrm{A}} & 0.2973 & 0.3696 & 0.5834 & 0.6604 & 0.8437 & 0.9931 \\ P_{\Lambda} / \mathrm{kPa} & 3.31 & 3.83 & 4.84 & 5.36 & 6.76 & 7.29 \\ P_{5} / \mathrm{kPa} & 28.16 & 26.08 & 20.42 & 18.01 & 10.0 & 0.47 \end{array} $$

Raoult's law is a fundamental principle of physical chemistry that describes the behavior of solutions. Specifically, it relates to the vapor pressure of a solvent in a solution containing non-volatile solutes or to the components in an ideal mixture. The law is expressed with the equation:

\( P_A = x_A \times P_A^* \)

where \( P_A \) is the partial vapor pressure of the component A in the mixture, \( x_A \) is the mole fraction of A in the liquid phase, and \( P_A^* \) is the vapor pressure of the pure component A at the same temperature. Raoult's law assumes that the solution behaves ideally, meaning that the interactions between molecules are the same as in the pure substances and there are no changes in volume upon mixing.

In solutions where the components form an ideal mixture, Raoult's law accurately predicts the vapor pressure. However, most real solutions deviate from ideal behavior. To facilitate the understanding for a student, let's remember that this law is best applied to solutions where the solute does not significantly affect the solvent's properties, which is generally the case in solutions with a solute at a relatively low concentration. In the context of the exercise, the region where Raoult's law is obeyed will show a linear relationship between the mole fraction and vapor pressure in the plot, indicating that the solution behaves ideally in this composition range.

Henry's law is another key concept in understanding the behavior of gas-liquid solutions. It is especially important in the context of very dilute solutions. The law states that the amount of dissolved gas in a liquid is directly proportional to the partial pressure of that gas above the liquid, at a constant temperature. The mathematical representation of Henry's law is given by:

\( P_B = k_H \times x_B \)

Here, \( P_B \) represents the partial pressure of gas B in the vapor phase, \( k_H \) is Henry's law constant for the solute B in a particular solvent, and \( x_B \) is the mole fraction of B in the liquid phase. Unlike Raoult's law which is better for the solvent and ideal mixtures, Henry's law is most useful for describing the solvation of gases in liquids at low concentrations. In the exercise, where benzene (\text{B}) is considered a solute, Henry's law applies in the region where the mole fraction of benzene is low, and the plot of vapor pressure versus mole fraction will show direct proportionality, illustrating that the dissolved benzene is not significantly affecting the properties of the majority solvent.

Activity coefficients are a measure of how the behavior of a species in mixture deviates from their behavior in an ideal mixture due to interactions between the molecules. The concept of activity is akin to the 'effective' concentration and can diverge substantially from the actual concentration in non-ideal solutions. The activity coefficient, \( \text{\textgamma}_A \), is defined by the relationship:

\( a_A = x_A \times \text{\textgamma}_A \)

The activity of component A, \( a_A \), reflects its effective concentration in the mixture, with \( x_A \) being its actual mole fraction. For an ideal solution, \( \text{\textgamma}_A \) equals 1, indicating no deviation from ideality. In the exercise, the determination of activity coefficients allows students to account for non-ideal behavior in a solution by comparing the actual vapor pressure to that predicted by Raoult's law for a given component. Calculating these coefficients also lays the groundwork for understanding more complex solution behavior and is vital for the calculation of excess Gibbs energy.

Excess Gibbs energy, denoted as \( G^{ex} \), is a thermodynamic property that provides crucial insight into the non-ideality of a solution. It represents the amount of energy that is above what would be expected for an ideal solution at the same composition and temperature. The excess Gibbs energy is calculated by integrating the activity coefficients over the entire composition range:

\( G^{ex} = R T \bigg(x_A \times ln(\text{\textgamma}_A) + x_B \times ln(\text{\textgamma}_B)\bigg) \)

where \( R \) is the gas constant, \( T \) is the temperature in Kelvin, and \( x_A \) and \( x_B \) are the mole fractions of components A and B. The \( ln(\text{\textgamma}_A) \) and \( ln(\text{\textgamma}_B) \) terms account for the deviation of component activities from ideality. A positive value of \( G^{ex} \) indicates repulsive interactions, whereas a negative value suggests attractive interactions between the components in the mixture. For the exercise presented, calculating the excess Gibbs energy over the composition range provides a quantitative measure of the non-ideality of the benzene-acetic acid mixture and is a fundamental concept for advancing the understanding of solution chemistry.