Question

Suppose \(50.0 \mathrm{mL}\) of \(0.250 \mathrm{M} \mathrm{CoCl}_{2}\) solution is added to 25.0 mL of 0.350 \(M \mathrm{NiCl}_{2}\) solution. Calculate the concentration, in moles per liter, of each of the ions present after mixing. Assume that the volumes are additive.

Short answer

After mixing, the concentration of each ion is \([\mathrm{Co}^{2+}] = 0.167 \, \mathrm{M}\), \([\mathrm{Ni}^{2+}] = 0.117 \, \mathrm{M}\), and \([\mathrm{Cl}^-] = 0.567 \, \mathrm{M}\).

Step-by-step solution

Calculate the total volume of the mixture

: Since the volumes are additive, we find the total volume (\(V_{total}\)) of the mixture by summing up the volumes of both solutions: \( V_{total} = 50.0 \, \mathrm{mL} + 25.0 \, \mathrm{mL} = 75.0 \, \mathrm{mL} \)

Find the moles of each ion before mixing

: First, let's find the moles of each ion in the individual solutions. For \(\mathrm{CoCl}_{2}\), we have one \(\mathrm{Co}^{2+}\) ion and two \(\mathrm{Cl}^-\) ions for every one unit. Moles of \(\mathrm{Co}^{2+}\) ions in the \(\mathrm{CoCl}_{2}\) solution: \( n_{\mathrm{Co}^{2+}} = (0.250 \, \mathrm{M})(50.0 \, \mathrm{mL}) = 12.5 \, \mathrm{mmol} \) Moles of \(\mathrm{Cl}^-\) ions in the \(\mathrm{CoCl}_{2}\) solution: \( n_{\mathrm{Cl}^-} = 2 \times (0.250 \, \mathrm{M})(50.0 \, \mathrm{mL}) = 25.0 \, \mathrm{mmol} \) For \(\mathrm{NiCl}_{2}\), we have one \(\mathrm{Ni}^{2+}\) ion and two \(\mathrm{Cl}^-\) ions for every one unit. Moles of \(\mathrm{Ni}^{2+}\) ions in the \(\mathrm{NiCl}_{2}\) solution: \( n_{\mathrm{Ni}^{2+}} = (0.350 \, \mathrm{M})(25.0 \, \mathrm{mL}) = 8.75 \, \mathrm{mmol} \) Moles of \(\mathrm{Cl}^-\) ions in the \(\mathrm{NiCl}_{2}\) solution: \( n_{\mathrm{Cl}^-} = 2 \times (0.350 \, \mathrm{M})(25.0 \, \mathrm{mL}) = 17.5 \, \mathrm{mmol} \)

Calculate concentrations after mixing

: Now that we have the total moles of each ion before mixing, we can find their concentrations after mixing by dividing their moles by the total volume in liters. Concentration of \(\mathrm{Co}^{2+}\) ions after mixing: \[ [\mathrm{Co}^{2+}] = \frac{12.5 \, \mathrm{mmol}}{75.0 \, \mathrm{mL}} = \frac{12.5 \, \mathrm{mmol}}{0.075 \, \mathrm{L}} = 0.167 \, \mathrm{M} \] Concentration of \(\mathrm{Ni}^{2+}\) ions after mixing: \[ [\mathrm{Ni}^{2+}] = \frac{8.75 \, \mathrm{mmol}}{75.0 \, \mathrm{mL}} = \frac{8.75 \, \mathrm{mmol}}{0.075 \, \mathrm{L}} = 0.117 \, \mathrm{M} \] Concentration of \(\mathrm{Cl}^-\) ions after mixing: \[ [\mathrm{Cl}^-] = \frac{25.0 \, \mathrm{mmol} + 17.5 \, \mathrm{mmol}}{75.0 \, \mathrm{mL}} = \frac{42.5 \, \mathrm{mmol}}{0.075 \, \mathrm{L}} = 0.567 \, \mathrm{M} \] Thus, after mixing, the concentration of each ion is \([\mathrm{Co}^{2+}] = 0.167 \, \mathrm{M}\), \([\mathrm{Ni}^{2+}] = 0.117 \, \mathrm{M}\), and \([\mathrm{Cl}^-] = 0.567 \, \mathrm{M}\).

Key concepts

Solution Concentration

Understanding the concentration of a solution is crucial for many chemical calculations. Concentration refers to the amount of a substance, typically measured in moles, present in a certain volume of solution. It tells us the strength of the solution, or how much solute is dissolved in the solvent.

There are various ways to express concentration, but one of the most common in chemistry is molarity. It is especially important in the context of reactions, as it allows for the calculation of reactants and products. When it comes to mixed solutions, like in the exercise given, calculating the concentration of each ion post-mixing is fundamental for understanding the resulting chemical behavior.

Molarity

Molarity is a unit of concentration that is defined as the number of moles of solute per liter of solution. Symbolically, it is expressed as moles per liter (M). To calculate the molarity, one must divide the total moles of the solute by the total volume of the solution in liters. For instance, in our example,\[ Molarity \, of \, Co^{2+} \, ions = \frac{12.5 \, mmol}{0.075 \, L} = 0.167 \, M \]

Understanding molarity is essential because it allows for direct comparison between different solutions' concentrations and plays a pivotal role in stoichiometry and chemical equations.

Stoichiometry

Stoichiometry is a section of chemistry that involves calculating the quantities of reactants and products in chemical reactions. It is based on the conservation of mass and the concept of mole ratios as determined by balanced chemical equations.

When dealing with solutions, stoichiometry helps to predict the outcome of reactions to understand the formation of products and consumption of reactants. As displayed in the exercise, knowing the stoichiometry—1 mole of \(CoCl_2\) contains 1 mole of \(Co^{2+}\) ions and 2 moles of \(Cl^-\) ions—is key to finding the concentration of ions after mixing different solutions.

Chemical Solutions

In chemistry, solutions are homogeneous mixtures composed of two or more substances. A solute is a substance that is dissolved in a solvent. The solute can be solids, liquids, or gases, and solvent is typically a liquid.

Chemical solutions are at the heart of many analytical techniques and synthesis processes. Understanding their properties, such as concentration, is vital for laboratory work, pharmaceutical formulations, and industrial processes. In the exercise we're discussing, two chemical solutions of \(CoCl_2\) and \(NiCl_2\) are mixed, and we are tasked with the calculation of ion concentrations in the new mixture.

Additive Volumes

When combining two solutions, it's often assumed that the volumes are additive, meaning the total volume of the mixture is the sum of the individual volumes of the solutions before they were mixed. This principle is utilized in the provided problem to calculate the final volume of the mixture as \(75.0 \, mL\).

This assumption simplifies calculations and is generally valid for dilute solutions, where the volume occupied by the solute is negligible compared to the volume of the solvent. However, for concentrated solutions, or when dealing with solutions of substances with significant volume changes upon mixing, corrections might be necessary to accommodate non-additive volume changes.