Question

Some \(\mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7},\) with a mass of \(2.335 \mathrm{~g},\) is dissolved in enough water to make \(500 . \mathrm{mL}\) solution. (a) Calculate the molarity of the potassium dichromate. (b) Calculate the concentrations of the \(\mathrm{K}^{+}\) and \(\mathrm{Cr}_{2} \mathrm{O}_{7}^{2-}\) ions.

Short answer

The molarity of \( \mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7} \) is 0.01588 M; \( \mathrm{K}^{+} \) is 0.03176 M, \( \mathrm{Cr}_{2} \mathrm{O}_{7}^{2-} \) is 0.01588 M.

Step-by-step solution

Find Molar Mass of K2Cr2O7

Calculate the molar mass of potassium dichromate \( \mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7} \). The formula consists of 2 potassium (K), 2 chromium (Cr), and 7 oxygen (O) atoms. Using atomic masses: K = 39.1 g/mol, Cr = 52.0 g/mol, and O = 16.0 g/mol, calculate the molar mass as follows: \( 2 \times 39.1 + 2 \times 52.0 + 7 \times 16.0 = 294.2 \ \mathrm{g/mol} \).

Calculate Moles of K2Cr2O7

Find the number of moles of \( \mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7} \) using its mass and molar mass. Use the formula: \( \text{moles} = \frac{\text{mass}}{\text{molar mass}} \). Substitute the values: \( \frac{2.335 \ \mathrm{g}}{294.2 \ \mathrm{g/mol}} = 0.00794 \ \, \mathrm{mol} \).

Calculate Molarity of K2Cr2O7

Molarity (M) is calculated as \( \frac{\text{moles of solute}}{\text{volume of solution in liters}} \). We have 0.00794 moles of \( \mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7} \) and a solution volume of 500 mL, which is 0.500 L. Therefore, molarity is \( \frac{0.00794 \ \mathrm{mol}}{0.500 \ \mathrm{L}} = 0.01588 \ \, \mathrm{M} \).

Determine Ion Ratios

In the compound \( \mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7} \), for every one formula unit, there are two potassium ions (\(\mathrm{K}^+\)) and one dichromate ion (\(\mathrm{Cr}_{2} \mathrm{O}_{7}^{2-}\)).

Calculate Concentration of K+ Ions

Since there are 2 \( \mathrm{K}^{+} \) ions per formula unit of \( \mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7} \), multiply the molarity of \( \mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7} \) by 2 to obtain the molarity of \( \mathrm{K}^{+} \): \( 2 \times 0.01588 \ \, \mathrm{M} = 0.03176 \ \, \mathrm{M} \).

Calculate Concentration of Cr2O7^2- Ions

The concentration of \( \mathrm{Cr}_{2} \mathrm{O}_{7}^{2-} \) ions is the same as that of \( \mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7} \), which is 0.01588 M.

Key concepts

Potassium Dichromate

Potassium dichromate, chemically represented as \( \mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7} \), is a bright orange crystalline solid. It is widely used in laboratories and various industrial applications. In this compound, there are two potassium (\( \mathrm{K} \)) atoms, two chromium (\( \mathrm{Cr} \)) atoms, and seven oxygen (\( \mathrm{O} \)) atoms, which together give it a robust and stable molecular structure. The molecular mass of potassium dichromate plays a crucial role in calculating its concentration in solution. The molar mass of potassium dichromate is determined by adding the atomic masses of all its constituent atoms:

• Potassium: 2 atoms \( \times 39.1 \ \mathrm{g/mol} \)
• Chromium: 2 atoms \( \times 52.0 \ \mathrm{g/mol} \)
• Oxygen: 7 atoms \( \times 16.0 \ \mathrm{g/mol} \)

This results in a total molar mass of \( 294.2 \ \mathrm{g/mol} \). Understanding the composition and calculation of potassium dichromate's molar mass is essential for conducting further solution chemistry exercises.

Ionic Concentration

The concept of ionic concentration is vital when dealing with solutions like potassium dichromate, as it decomposes into ions when dissolved in water. Once in solution, it dissociates into potassium ions (\( \mathrm{K}^+ \)) and dichromate ions (\( \mathrm{Cr}_{2} \mathrm{O}_{7}^{2-}\)). It contains two \( \mathrm{K}^+ \) ions and one \( \mathrm{Cr}_{2} \mathrm{O}_{7}^{2-} \) ion per formula unit of \( \mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7} \). The concentration of ions in a solution can be calculated by considering the stoichiometry of the salt in relation to its dissociation. For example:- If the molarity of the potassium dichromate solution is \( 0.01588 \ \mathrm{M} \), then using the ion-to-molecule ratio, the concentration of \( \mathrm{K}^+ \) ions becomes \( 2 \times 0.01588 \ \mathrm{M} = 0.03176 \ \mathrm{M} \).- Meanwhile, the concentration of \( \mathrm{Cr}_{2} \mathrm{O}_{7}^{2-} \) ions remains \( 0.01588 \ \mathrm{M} \).Understanding how to derive these concentrations from molarity helps outline the behavior of ionic compounds in solutions, which is pivotal in solution chemistry.

Solution Chemistry

Solution chemistry is a field that focuses on the properties and behaviors of solutes and solvents when they are mixed. Molarity, a key concept in this field, helps quantify the concentration of a solute in a solution. The calculation of molarity involves determining the number of moles of solute and dividing it by the volume of the solution in liters. For instance, the molarity of potassium dichromate when \( 2.335 \ \mathrm{g} \) is dissolved in \( 500 \ \mathrm{mL} \) of water is calculated as follows:1. Determine moles of solute using the formula: \[ \text{moles} = \frac{\text{mass}}{\text{molar mass}} \]2. Moles of \( \mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7} \) are calculated to be \( 0.00794 \ \mathrm{mol} \).3. Convert the volume of the solution to liters, here \( 500 \ \mathrm{mL} = 0.500 \ \mathrm{L} \).4. Calculate molarity: \[ \text{Molarity} = \frac{0.00794 \ \mathrm{mol}}{0.500 \ \mathrm{L}} = 0.01588 \ \mathrm{M} \]These calculations provide a clear representation of how solute quantity influences the solution's composition and its chemical behavior, key to understanding solution chemistry.