Question

A mathematical equation useful for dilution calculations is \(M_{\text {dil }} \times V_{\text {dil }}=M_{\text {cunc }} \times V_{\text {cvac }}\). (a) What does each symbol mean, and why does the equation work? (b) Given the volume and molarity of a \(\mathrm{CaCl}_{2}\) solution, how do you determine the amount (mol) and the mass (g) of solute?

Short answer

The equation relates molarity and volume of solutions. To find moles: molarity \times volume. To find mass: moles \times molar mass.

Step-by-step solution

Understanding the Equation

The equation given is \( M_{\text {dil }} \times V_{\text {dil }} = M_{\text {conc }} \times V_{\text {conc }} \). Each symbol represents a specific quantity: - \( M_{\text {dil }} \) is the molarity of the diluted solution. - \( V_{\text {dil }} \) is the volume of the diluted solution. - \( M_{\text {conc }} \) is the molarity of the concentrated solution. - \( V_{\text {conc }} \) is the volume of the concentrated solution. This equation works because it follows the principle of the conservation of moles of solute during the dilution process.

Using the Equation to Determine Amount of Solute

To determine the amount of solute (in moles) present in a solution given its volume and molarity, use the formula: \[ \text{Moles of solute} = M \times V \] where \( M \) is in mol/L and \( V \) is in liters.

Calculating Moles of \( \text{CaCl}_2 \)

Given the volume \( V \) and molarity \( M \) of a \( \text{CaCl}_2 \) solution, plug these values into the moles formula: \[ \text{Moles of } \text{CaCl}_2 = M \times V \]

Converting Moles to Grams

To convert the amount in moles to mass (grams), multiply the number of moles by the molar mass of the solute. For \( \text{CaCl}_2 \), the molar mass is the sum of the atomic masses of calcium (Ca) and chlorine (Cl): \[ M_{\text{CaCl}_2} = 40.08 + (2 \times 35.45) = 110.98 \text{ g/mol} \] So, the mass of \( \text{CaCl}_2 \) in grams can be calculated as: \[ \text{Mass of } \text{CaCl}_2 = \text{Moles of } \text{CaCl}_2 \times 110.98\]

Applying to a Specific Example

Suppose you have a 0.5 M solution of \( \text{CaCl}_2 \) and the volume is 2 L. The amount of \( \text{CaCl}_2 \) in moles would be: \[ \text{Moles of } \text{CaCl}_2 = 0.5 \text{ M} \times 2 \text{ L} = 1 \text{ mole} \] Then, converting this to grams: \[ \text{Mass of } \text{CaCl}_2 = 1 \text{ mole} \times 110.98 \text{ g/mol} = 110.98 \text{ grams} \]

Key concepts

headline of the respective core concept

Dilution calculations are a common task in chemistry. The key formula used is the dilution equation: \( M_{\text {dil }} \times V_{\text {dil }} = M_{\text {conc }} \times V_{\text {conc }} \). This helps us determine how to achieve a desired concentration of a solution by adding solvent to a more concentrated one. Let's break down what each term represents and how you can apply this equation effectively.
\( M_{\text {dil }} \) stands for the molarity (concentration) of the final diluted solution, while \( V_{\text {dil }} \) is its volume.
\( M_{\text {conc }} \) is the molarity of the initial concentrated solution, and \( V_{\text {conc }} \) is the volume taken from this concentrated solution. This relationship ensures that the number of moles of solute remains constant during dilution.

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Molarity (\( M \)) is a measure of concentration in chemistry, defined as the number of moles of solute per liter of solution (\( mol/L \)). It’s a core concept because it helps us quantify how much solute is present in a given volume of solution. Imagine you have a glass of water with some salt dissolved in it. The molarity tells us how 'salty' the water is by specifying the amount of salt (solute) in each liter of water (solvent). For example, a 1 M (\( M \)) solution has 1 mole of solute in every liter of solution.
To calculate molarity, use the formula:\[ M = \frac{\text{moles of solute}}{\text{liters of solution}} \]This is crucial for preparing solutions with precise concentrations in scientific experiments.

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The principle of conservation of moles ensures the number of moles of solute remains unchanged during dilution. When you dilute a solution, you add more solvent but do not change the total amount of solute. For instance, if you dilute 1 liter of 1M NaCl solution to 2 liters, the molarity (concentration) of the solution changes, but the number of moles of NaCl remains the same.This principle is mathematically represented by:\( M_{\text{dil}} \times V_{\text{dil}} = M_{\text{conc}} \times V_{\text{conc}} \). Here’s why it works: when you multiply molarity (\( M \)) by volume (\( V \)), you get the number of moles. So, basically, the equation balances out the number of moles before and after dilution, making sure nothing is lost or gained, just spread out more or less.

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When you want to find the mass of a solute in a solution, you often start by calculating the moles of the solute and then convert those moles to mass. Here’s a step-by-step breakdown:
1. First, determine the number of moles of solute using the formula:\[ \text{Moles of solute} = M \times V \]where \( M \) is molarity and \( V \) is volume in liters.
2. Next, convert moles to grams using the molar mass of the solute. For example, for calcium chloride (\( CaCl_2 \)), the molar mass is:\[ M_{\text{CaCl}_2} = 40.08 + (2 \times 35.45) = 110.98 \text{ g/mol} \]Finally, use this conversion formula:\[ \text{Mass of solute} = \text{Moles of solute} \times \text{Molar mass} \]For instance, if you have 1 mole of \( CaCl_2 \), the mass in grams would be:\[ \text{Mass of } \text{CaCl}_2 = 1 \text{ mole} \times 110.98 \text{ g/mol} = 110.98 \text{ grams} \]This process allows you to translate between concentrations and actual amounts, which is essential for practical lab work.