Question

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

Short answer

The concentration of the final solution is 1.093 M.

Step-by-step solution

Calculate Moles in First Solution

To find the moles of Ca(NO3)2 in the first solution, use the formula: \[ ext{moles} = ext{volume (L)} \times ext{concentration (M)} \]Convert 46.2 mL to liters (0.0462 L). Then, calculate the moles:\[ ext{moles from first solution} = 0.0462 ext{ L} \times 0.568 ext{ M} = 0.02625 ext{ moles} \]

Calculate Moles in Second Solution

Convert 80.5 mL to liters (0.0805 L). Then use the same formula for the second solution:\[ ext{moles from second solution} = 0.0805 ext{ L} \times 1.396 ext{ M} = 0.11238 ext{ moles} \]

Calculate Total Moles

Add the moles from both solutions to get the total moles:\[ ext{total moles} = 0.02625 ext{ moles} + 0.11238 ext{ moles} = 0.13863 ext{ moles} \]

Calculate Total Volume

Add the volumes of both solutions in liters to get the total volume:\[ ext{total volume} = 0.0462 ext{ L} + 0.0805 ext{ L} = 0.1267 ext{ L} \]

Calculate Final Concentration

Use the formula for concentration: \[ ext{concentration} = \frac{ ext{total moles}}{ ext{total volume (L)}} \]Substitute the known values:\[ ext{final concentration} = \frac{0.13863 ext{ moles}}{0.1267 ext{ L}} = 1.093 ext{ M} \]

Key concepts

Molarity

Molarity is a way to express the concentration of a solute in a solution. It's defined as the number of moles of solute divided by the volume of solution in liters. This measurement is widely used in chemistry because it provides a simple way to understand how concentrated a solution is. The formula for molarity is:\[ M = \frac{\text{moles of solute}}{\text{liters of solution}} \]
Understanding molarity helps you know how much of a chemical, like calcium nitrate, is present in a given volume. For example, if you have a 0.568 M solution of calcium nitrate, it means there are 0.568 moles of calcium nitrate in every liter of that solution. Molarity is crucial when preparing solutions, reacting chemicals precisely, or studying their behaviors under different conditions.

Calcium Nitrate

Calcium nitrate, with the chemical formula \(\text{Ca(NO}_3\text{)}_2\), is a compound that appears as colorless crystals. It is an inorganic nitrate salt of calcium and is mainly used in fertilizers, as it provides essential nutrients to plants. Calcium nitrate supplies calcium and nitrate, which are vital for plant growth and development.
In chemistry problems, calcium nitrate is used to illustrate the concept of solution concentration due to its solubility in water and significance in agricultural practices. When dissolved, it dissociates into calcium ions and nitrate ions, facilitating various reactions and experiments. Understanding its behavior in solutions helps in predicting chemical reactions and interactions.

Moles Calculation

The concept of moles is central to understanding chemistry. A mole is a unit that quantifies an amount of substance. One mole contains exactly \(6.022 \times 10^{23}\) particles, be it atoms, molecules, or ions, known as Avogadro's number.
To calculate moles in a solution, you use the formula:\[ \text{moles} = \text{volume (L)} \times \text{concentration (M)} \]
In the example provided, first, convert the solution volume from milliliters to liters by dividing by 1000. Then multiply by the molarity to find the moles present in that solution. This step is key for understanding how much of a solute (like calcium nitrate) is present in a specific volume, which is essential for predicting and analyzing chemical reactions.

Dilution

Dilution is a process of reducing the concentration of a solute in a solution, typically by mixing it with more solvent. This is important when solutions are too concentrated for a given experimental or operational requirement.
Dilution can be calculated using the formula: \[ C_1V_1 = C_2V_2 \]
Where \( C_1\) and \( V_1\) represent the concentration and volume of the initial solution, while \( C_2\) and \( V_2\) are those of the diluted solution. It helps maintain control over the strength of the solution for experimental consistency. When combining different solutions, as in the exercise above, dilution plays a role in determining the final solution's concentration, ensuring that reactions proceed as desired without exceeding safe or functional concentration limits.