Question

Acetonitrile $\left({\mathrm{CH}}_3\mathrm{CN}\right)$ is a polar organic solvent that dissolves a wide range of solutes, including many salts. The density of a $1.80\mathrm{M}$ LiBr solution in acetonitrile is $0.826\mathrm{g}/{\mathrm{cm}}^3$. Calculate the concentration of the solution in (a) molality, (b) mole fraction of LiBr, (c) mass percentage of ${\mathrm{CH}}_3\mathrm{CN}$.

Short answer

The concentration of the LiBr solution in acetonitrile is: (a) 2.94 mol/kg in molality, (b) 0.1086 in mole fraction of LiBr, and (c) 73.56% in mass percentage of $C{H}_{3}CN$.

Step-by-step solution

Calculate the molar mass of LiBr and Acetonitrile

First, we need to calculate the molar mass of LiBr and acetonitrile. LiBr: Li=6.939 g/mol, Br=79.904 g/mol, so LiBr = 6.939 + 79.904 = 86.843 g/mol Acetonitrile: C=12.01 g/mol, H=1.008 g/mol, N=14.007 g/mol, so CH3CN = 12.01 + 3*1.008 + 14.007 = 41.053 g/mol

Calculate the volume and mass of 1L solution

Now, we must determine the volume and mass of a 1-liter solution. Density = 0.826 g/cm³, so the mass of 1000 cm³ (1L) = density * volume = 0.826 g/cm³ * 1000 cm³ = 826 g

Calculate the moles of LiBr and Acetonitrile in 1L of solution

Next, we need to find the moles of LiBr and acetonitrile in 1L of solution. Molarity of LiBr = 1.80 mol/L Moles of LiBr in 1L solution = 1.80 mol Moles of Acetonitrile in 1L solution = (Mass of solution - Moles of LiBr × Molar mass of LiBr) / Molar mass of Acetonitrile Moles of Acetonitrile = (826 g - (1.80 mol × 86.843 g/mol)) / 41.053 g/mol ≈ 14.76 mol

Calculate the molality of the solution

Now we can calculate the molality of the solution (m), which is defined as moles of solute (LiBr) per kilogram of solvent (Acetonitrile). Molality = (moles of LiBr) / (mass of CH3CN in kg) Molality = (1.80 mol)/(14.76 mol * 41.053 g/mol * 0.001 kg/g) ≈ 2.94 mol/kg

Calculate the mole fraction of LiBr

Next, we calculate the mole fraction of LiBr (X_LiBr), which is the ratio of moles of LiBr to the total moles in the solution. Mole fraction of LiBr = (moles of LiBr) / (moles of LiBr + moles of Acetonitrile) X_LiBr = (1.80 mol)/(1.80 mol + 14.76 mol) ≈ 0.1086

Calculate the mass percentage of Acetonitrile

Finally, we need to calculate the mass percentage of acetonitrile in the solution, which is the ratio of the mass of acetonitrile to the total mass of the solution, expressed as a percentage. Mass percentage of Acetonitrile = (mass of Acetonitrile) / (mass of solution) * 100 Mass percentage of Acetonitrile = (14.76 mol * 41.053 g/mol) / (826 g) * 100 ≈ 73.56% The calculated values are: (a) Molality: 2.94 mol/kg (b) Mole fraction of LiBr: 0.1086 (c) Mass percentage of Acetonitrile: 73.56%

Key concepts

Acetonitrile

Acetonitrile, with the chemical formula ${\text{CH}}_3\text{CN}$, is a versatile organic solvent widely used in the lab for various applications. It’s known for its ability to dissolve a broad spectrum of solutes, including many salts. This makes it an important solvent in chemical reactions and chromatography. The polar nature of acetonitrile, due to the presence of a nitrile group, allows it to stabilize ions and interact effectively with different substances.
Despite its usefulness, acetonitrile should be handled with care due to its flammable nature. In the context of solution concentration calculations, knowing its molar mass and density is crucial to determine the concentration forms like molality, mole fraction, and mass percentage.

Molality

Molality is a concentration unit expressed as the number of moles of solute per kilogram of solvent. Unlike molarity, it does not depend on volume, which varies with temperature, making molality a more stable concentration measure under varying conditions.
To calculate molality in an acetonitrile solution, it's essential to first determine the moles of the solute, such as LiBr, and the mass of the solvent, which in this case is acetonitrile. The basic formula is:

• Molality $(m)=\frac{\text{moles of solute}}{\text{kilograms of solvent}}$

This allows for precise calculations, especially when dealing with temperature-sensitive experiments.

Mole Fraction

The mole fraction is a way of expressing the composition of a mixture. It is the ratio of the moles of one component to the total moles in the solution. When dealing with a solution of LiBr in acetonitrile, the mole fraction is calculated by taking the moles of LiBr divided by the sum of moles of both LiBr and acetonitrile.
The formula is:

• Mole fraction of LiBr $(X_{LiBr})=\frac{\text{moles of LiBr}}{\text{moles of LiBr}+\text{moles of acetonitrile}}$

This unitless measurement provides a proportionate insight into the mixture, crucial for understanding the system's chemical properties.

Mass Percentage

Mass percentage is a way to express concentration by considering the mass of a particular component in relation to the total mass of the mixture, and then multiplying by 100 to get a percentage. For acetonitrile in a LiBr solution, this means calculating what percentage of the total solution's mass is due to the acetonitrile.
The formula for calculating mass percentage is:

• Mass \% of Acetonitrile = $\frac{\text{mass of acetonitrile}}{\text{total mass of solution}}\times 100$

It’s a straightforward way to compare the mass contributions of different components, especially useful in quality control and formulation tasks.