Problem 59 - Chemistry and Chemical Reactivity

Question

Concentrated aqueous ammonia has a molarity of \(14.8 \mathrm{mol} / \mathrm{L}\) and a density of \(0.90 \mathrm{g} / \mathrm{cm}^{3} .\) What is the molality of the solution? Calculate the mole fraction and weight percent of \(\mathrm{NH}_{3}\)

Step-by-Step Solution

Answer

Molality is 22.8 mol/kg; mole fraction is 0.291; weight percent is 27.9%.

Step 1: Define Terms and Given Values

We are given the following:- Molarity of ammonia solution, \(M = 14.8 \, \text{mol/L}\)- Density of the solution, \(\rho = 0.90 \, \text{g/cm}^3\)We need to find the molality, mole fraction, and weight percent of \(\text{NH}_3\) in the solution.

Step 2: Convert Density to Units Useful for Molality

Given density \(0.90 \, \text{g/cm}^3\) is equivalent to \(0.90 \, \text{g/mL}\) or \(900 \, \text{g/L}\). This means each liter of solution weighs 900 grams.

Step 3: Calculate Mass of Ammonia in One Liter of Solution

Using the molarity, \(14.8 \, \text{mol/L}\), the number of moles of \(\text{NH}_3\) in 1 liter is 14.8 mol. The molar mass of \(\text{NH}_3\) is approximately 17.03 \(\text{g/mol}\). Mass of ammonia \(= 14.8 \, \text{mol} \times 17.03 \, \text{g/mol} = 251.044 \, \text{g}\).

Step 4: Calculate Mass of Water in Solution

The mass of the total solution is 900 g for 1 liter. Thus, the mass of water \(= 900 \, \text{g} - 251.044 \, \text{g} = 648.956 \, \text{g}\).

Step 5: Calculate Molality of the Solution

Molality \(m\) is defined as moles of solute per kilogram of solvent.\[ m = \frac{14.8 \, \text{mol NH}_3}{0.648956 \, \text{kg H}_2\text{O}} = 22.8 \, \text{mol/kg} \]

Step 6: Calculate Mole Fraction of Ammonia

To find the mole fraction, calculate moles of water: \(\frac{648.956 \, \text{g}}{18.015 \, \text{g/mol}} = 36.036 \, \text{mol}\).The mole fraction of \(\text{NH}_3\) is:\[ \text{Mole fraction of } \text{NH}_3 = \frac{14.8}{14.8 + 36.036} = 0.291 \]

Step 7: Calculate Weight Percent of Ammonia

Weight percent is defined as the mass fraction of a component in a mixture. For ammonia:\[ \text{Weight percent} = \left( \frac{251.044 \, \text{g}}{900 \, \text{g}} \right) \times 100\% = 27.9\% \]

Key Concepts

Molarity

Molarity is a measure of the concentration of a solute in a solution. It is defined as the number of moles of solute per liter of solution. This concept is crucial for understanding how concentrated a solution is.

In our exercise, we have a molarity of ammonia (\( M \)) which is given as 14.8 mol/L. This value tells us that there are 14.8 moles of ammonia in every liter of the solution.

To use molarity effectively, especially in calculations like molality or weight percent, it’s important to start by determining the moles of solute using the molarity, which can then be used to calculate other properties of the solution.

Density Conversion

Density is another key concept in solution calculations. It is the mass of a substance divided by its volume. In this exercise, the density of the solution is 0.90 g/cm³. Since 1 cm³ is equal to 1 mL, this shows us that our density can also be seen as 0.90 g/mL.

In solution calculations, converting density into different units helps in subsequent calculations. Therefore, converting it to g/L, which becomes 900 g/L, gives us the mass of the solution for every liter. This conversion is essential to find out the mass of the solution, allowing us to deduce the mass of the solvent by subtraction.

• Knowing this, we can calculate the mass of the entire solution and therefore move on to calculate specifics such as molality and weight percent.

Mass Calculation

Mass calculation in solution chemistry involves determining the mass of different components within a solution. In our example, the mass of ammonia (NH₃) within the solution can be calculated using its molarity and molar mass.

Using the molarity of 14.8 mol/L and the molar mass of ammonia, roughly 17.03 g/mol, you calculate the mass of NH₃ in one liter of solution. Here’s the formula: \[ \text{Mass of NH}_3 = 14.8 \, \text{mol} \times 17.03 \, \text{g/mol} = 251.044 \, \text{g}\]
This gives us the mass of the solute (NH₃) in the solution, which is critical for finding the mass of the solvent and ultimately other concentration measures like molality.

Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a mixture. It is defined as the ratio of moles of a specific component to the total moles of all components in the mixture.

In this exercise, to find the mole fraction of ammonia, you also need to calculate the moles of the solvent—water. With the solution's mass of water being 648.956 g, and water's molar mass approximately 18.015 g/mol, you find the moles of water. Then,

1. Total moles = moles of NH₃ + moles of water
2. Mole fraction = moles of NH₃ / total moles

Hence, the mole fraction provides insight into the composition of a solution on a molecular level, showing proportional amounts of each component.

Weight Percent

Weight percent is often used to describe the concentration of a solute in a solution. It's defined as the mass of the solute divided by the total mass of the solution, multiplied by 100% to convert it into a percentage.

In our scenario, once we calculate the mass of ammonia (NH₃) to be 251.044 g and the total solution mass as 900 g, we can determine the weight percent:

• Weight Percent = (Mass of Ammonia / Total Mass of Solution) × 100%
• = (251.044 g / 900 g) × 100% = 27.9%

This percentage helps us understand the proportion of NH₃ by weight within the whole solution, giving a clear idea of how concentrated the solution is in practical, everyday terms.