Question

A \(46.2-\mathrm{mL}, 0.568 M\) calcium nitrate \(\left[\mathrm{Ca}\left(\mathrm{NO}_{3}\right)_{2}\right]\) solution is mixed with \(80.5 \mathrm{~mL}\) of \(1.396 \mathrm{M}\) calcium nitrate solution. Calculate the concentration of the final solution.

Short answer

The concentration of the final solution is \(1.094 M\).

Step-by-step solution

Calculate the total moles of calcium nitrate in each solution

Use the formula \(M = n/V\) to find the moles of calcium nitrate in each solution, where \(M\) is molarity, \(n\) is the number of moles, and \(V\) is the volume. Therefore, for the first solution, the moles are \(n_1 = M_1V_1 = 0.568 M * 46.2 mL = 26.22 mmol\), and for the second solution, the moles are \(n_2 = M_2V_2 = 1.396 M * 80.5 mL = 112.38 mmol\). Note that \(M = mol/L\) and that the volumes are in mL, so the number of moles is in mmol.

Determine the total moles of calcium nitrate

Combine the moles of calcium nitrate in both solutions to get the total moles. So, \(n_t = n_1 + n_2 = 26.22 mmol + 112.38 mmol = 138.6 mmol\).

Find the total volume of the mixed solutions

Add the volumes of the initial solutions to get the total volume. Hence, \(V_t = V_1 + V_2 = 46.2 mL + 80.5mL = 126.7 mL\).

Calculate the final concentration

Use the formula for molarity, \(M = n/V\), to get the final concentration. Substitute the total moles and the total volume to get the final concentration. Therefore, \(M_f = n_t/V_t = 138.6 mmol / 126.7 mL = 1.094 M\).

Key concepts

Molarity Formula

Understanding the molarity formula is crucial for calculating the concentration of solutions. Molarity, denoted by the symbol 'M,' is a measure of concentration that tells us how many moles of a solute are present in a liter of solution. The formula to calculate molarity is given as:
\[ M = \frac{n}{V} \]
where 'M' is the molarity, 'n' represents the number of moles of the solute, and 'V' is the volume of the solution in liters. When volume is expressed in milliliters (mL), we can still use this formula by converting mL to L, where 1 L = 1000 mL.
To calculate the moles, you need to know the quantity of the substance in grams and its molar mass. After finding the number of moles, you simply divide by the volume of the solution to find the molarity. This formula is pivotal for mixing solutions to achieve desired concentrations, as seen in our exercise problem where two calcium nitrate solutions with different molarities are mixed.

Concentration of Solutions

The concentration of a solution is a measure of how much solute is dissolved in a specific amount of solvent. There are various ways to express concentrations, but molarity is one of the most common methods used in chemistry due to its straightforward relationship with the volume of the solution.
Molarity is especially helpful because it allows chemists to easily calculate the amount of a substance in a solution, necessary for reactions and dilutions. When two solutions with known molarities are mixed, the concentration of the final solution can be found by adding the moles of solute from both original solutions and dividing by the total volume, as demonstrated in the provided solution.
This concept is not just limited to academic exercises but also essential in real-life applications, such as formulating medicines, calibrating pH solutions, or creating standard solutions for laboratory experiments.

Moles and Volume Relationship

The relationship between moles and volume is a fundamental aspect of chemical solutions. The number of moles (\(n\)) is a measure of the quantity of particles (atoms, molecules, ions, etc.) and can be derived from the mass of the substance and its molar mass. Volume (\(V\)), particularly in the context of solutions, is typically measured in liters or milliliters.
When calculating the final concentration of a mixed solution, understanding this relationship is essential. It's all about balancing the number of moles with the resultant volume. For the individual components before mixing, the molarity (moles per volume) gives us a clear picture of their concentrations. After mixing, it's the total number of moles divided by the total volume that yields the final concentration.
Using the problem at hand, the exercises' advised approach in combining the moles from both solutions and then the total volume highlights how to manipulate these quantities to arrive at a consolidated concentration for a new, combined solution.