Problem 57 - Chemistry and Chemical Reactivity

Question

Dimethylglyoxime \(\left[\mathrm{DMG},\left(\mathrm{CH}_{3} \mathrm{CNOH}\right)_{2}\right]\) is used as a reagent to precipitate nickel ion. Assume that \(53.0 \mathrm{g}\) of DMG has been dissolved in \(525 \mathrm{g}\) of ethanol \(\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\right)\) (IMAGE CANNOT COPY) (a) What is the mole fraction of DMG? (b) What is the molality of the solution? (c) What is the vapor pressure of the ethanol over the solution at ethanol's normal boiling point of 78.4 ^ C? (d) What is the boiling point of the solution? (DMG does not produce ions in solution.) \(\left(K_{\mathrm{bn}} \text { for ethanol }=+1.22^{\circ} \mathrm{C} / \mathrm{m}\right)\)

Step-by-Step Solution

Answer

(a) 0.0386; (b) 0.869 m; (c) 0.9614 atm; (d) 79.46°C.

Step 1: Calculate Moles of DMG

First, find the molar mass of DMG,\( \text{(CH}_3\text{CNOH)}_2 \). Carbon (C) has a molar mass of 12.01 g/mol, nitrogen (N) 14.01 g/mol, oxygen (O) 16.00 g/mol, and hydrogen (H) 1.01 g/mol. \[ \text{C}_4\text{H}_8\text{N}_2\text{O}_2 = 12.01 \times 4 + 1.01 \times 8 + 14.01 \times 2 + 16.00 \times 2 = 116.12 \text{ g/mol} \]. Now calculate the moles of DMG using its mass: \( \frac{53.0 \text{ g}}{116.12 \text{ g/mol}} = 0.4567 \text{ moles} \).

Step 2: Calculate Moles of Ethanol

Find the molar mass of ethanol, \( \text{C}_2\text{H}_5\text{OH} \), which is \( 12.01 \times 2 + 1.01 \times 6 + 16.00 = 46.08 \text{ g/mol} \). Calculate the moles of ethanol: \( \frac{525 \text{ g}}{46.08 \text{ g/mol}} = 11.39 \text{ moles} \).

Step 3: Calculate Mole Fraction of DMG

Mole fraction is calculated by dividing the moles of DMG by the total moles in the solution. \( x_{\text{DMG}} = \frac{0.4567}{0.4567 + 11.39} = 0.0386 \).

Step 4: Calculate Molality of the Solution

Molality is moles of solute per kilogram of solvent. \( m = \frac{0.4567 \text{ moles}}{0.525 \text{ kg}} = 0.869 \text{ m} \).

Step 5: Vapor Pressure of Ethanol over the Solution

Use Raoult's law: \( P = x_{\text{solvent}} \times P^{\circ} \). The mole fraction of ethanol \( x_{\text{ethanol}} = 1 - 0.0386 = 0.9614 \). At the boiling point, the vapor pressure equals 1 atm, so \( P = 0.9614 \times 1 \text{ atm} = 0.9614 \text{ atm} \).

Step 6: Calculate Boiling Point Elevation

Boiling point elevation \( \Delta T_b = K_b \times m \). \( \Delta T_b = 1.22^{\circ} \text{C/m} \times 0.869 \text{ m} = 1.06^{\circ} \text{C} \).

Step 7: Calculate New Boiling Point

Add the boiling point elevation to the normal boiling point of ethanol to get the new boiling point: \( 78.4^{\circ} \text{C} + 1.06^{\circ} \text{C} = 79.46^{\circ} \text{C} \).

Key Concepts

Mole Fraction Calculation

Understanding the concept of mole fraction is essential in solution chemistry. It is a way of expressing the concentration of a component in a mixture and is particularly useful when dealing with solutions. The mole fraction is defined as the ratio of the number of moles of a component to the total number of moles of all components in the solution. In our example involving dimethylglyoxime (DMG) and ethanol, we first calculated the moles of each component using their respective masses and molar masses. Once we know these values, the mole fraction of DMG is calculated by:

• Dividing the moles of DMG by the sum of the moles of DMG and ethanol.

In simpler terms, the mole fraction formula is: \[ x_{\text{DMG}} = \frac{\text{Moles of DMG}}{\text{Total moles}} \] This provides a number that represents the proportion of DMG in the solution. Understanding mole fractions helps us to predict how much each substance will contribute to the properties of the solution.

Molality Determination

Molality is another important concentration unit used in chemistry, especially when solutions undergo changes in temperature or pressure. It's based on the moles of solute per kilogram of solvent, which gives it stability unaffected by temperature changes, unlike molarity. In our exercise, after calculating the moles of DMG and knowing the mass of ethanol (the solvent), we find molality using:

• Molality \(m\) = \(\frac{\text{Moles of solute}}{\text{Kilograms of solvent}}\)

For DMG in ethanol, using the previously calculated moles and the mass of ethanol converted into kilograms, we find a molality of 0.869 mol/kg. Molality is particularly useful in scenarios like these where temperature changes can affect the volume and thus the molarity of the solution.

Raoult's Law

Raoult's Law is fundamental in understanding the behavior of ideal solutions, particularly in predicting vapor pressures. It states that the partial vapor pressure of each component in a solution is directly proportional to its mole fraction. This law becomes very useful when determining how the addition of a solute affects a solvent's vapor pressure.For our example, Raoult's Law is utilized to find the vapor pressure exerted by ethanol over the solution at its normal boiling point. The mole fraction of ethanol is calculated as:

• \(x_{\text{ethanol}} = 1 - x_{\text{DMG}}\)

The actual vapor pressure is then given by the product of this mole fraction and the pure solvent's vapor pressure, represented as:\[ P = x_{\text{solvent}} \times P^{\circ} \]Where \(P^{\circ}\) is the vapor pressure of the pure solvent. In practice, the calculated vapor pressure helps us understand how the presence of a solute such as DMG minimizes ethanol's vapor pressure.

Boiling Point Elevation

Boiling point elevation is a colligative property, meaning it depends on the number of solute particles in a solution, rather than their individual properties. When a solute is dissolved in a solvent, the solution's boiling point increases because the solute decreases the solvent's vapor pressure, requiring higher temperatures to reach the boiling point.The change in boiling point can be calculated using the formula:

• \(\Delta T_b = K_b \times m\)

Here, \(\Delta T_b\) represents the boiling point elevation, \(K_b\) is the ebullioscopic constant of the solvent (given as 1.22°C/m for ethanol), and \(m\) is the molality of the solution. In the case of our DMG and ethanol solution, this formula allows calculating a boiling point elevation of 1.06°C, resulting in a new boiling point of 79.46°C from the original boiling point of ethanol's 78.4°C. Understanding how solutes affect boiling points is critical in many practical applications, such as designing antifreeze solutions.