Problem 77 - Chemistry Matter and Change

Question

Experiment In the lab, you dilute 55 \(\mathrm{mL}\) of a 4.0 \(\mathrm{M}\) solution to make 250 \(\mathrm{mL}\) of solution. Calculate the molarity of the new solution.

Step-by-Step Solution

Answer

The molarity of the new solution is 0.88 M.

Step 1: Understand the Initial Setup

We begin with 55 mL of a 4.0 M solution. This means we have a concentrated solution and are adding a solvent to dilute it.

Step 2: Introduction to Molarity and Dilution Formula

We use the dilution formula, which is: \( M_1 \cdot V_1 = M_2 \cdot V_2 \). Here, \( M_1 \) is the initial molarity (4.0 M), \( V_1 \) is the initial volume (55 mL), and \( V_2 \) is the final volume (250 mL). We need to find the new molarity, \( M_2 \).

Step 3: Input Known Values into the Formula

Substitute the known values into the formula: \( 4.0 \cdot 55 = M_2 \cdot 250 \).

Step 4: Solve for New Molarity \( M_2 \)

Rearrange the formula to solve for \( M_2 \):\[ M_2 = \frac{4.0 \times 55}{250} \].Calculate \( M_2 \).

Step 5: Complete the Calculation

Perform the calculation:\[ M_2 = \frac{220}{250} = 0.88 \ \text{M} \]This is the molarity of the new solution.

Key Concepts

Molarity

Molarity is a measure of concentration that tells us how many moles of a solute are present in one liter of solution. It is represented by the symbol "M." When dealing with molarity, imagine how packed full of solute particles your solution is. The more solute, the higher the molarity. To calculate molarity, use the formula: \[ \text{Molarity (M)} = \frac{\text{moles of solute}}{\text{liters of solution}} \]This formula helps quantify exactly how concentrated a solution is, which is crucial in preparing solutions for experiments or reactions. Knowing the molarity allows you to predict how a solution might behave or react when used in different scenarios.

Dilution Formula

When you dilute a solution, you add more solvent without adding more solute, which decreases its concentration. The dilution formula, \( M_1 \cdot V_1 = M_2 \cdot V_2 \), is a handy tool for calculating how the molarity changes when you dilute a solution. Here's a breakdown of what each symbol means:

• \( M_1 \): Initial molarity, or the concentration of the starting solution.
• \( V_1 \): Initial volume of the concentrated solution.
• \( M_2 \): Final molarity, or the concentration after dilution.
• \( V_2 \): Final volume of the diluted solution.

This formula is based on the principle that the amount of solute remains the same before and after dilution. It effectively captures the relationship between the concentrations and volumes before and after dilution.

Calculation Steps

The process of calculating the new molarity after dilution is quite straightforward if you follow the steps carefully.Firstly, start by identifying your known values. In this example, the initial molarity \( M_1 \) is 4.0 M, the initial volume \( V_1 \) is 55 mL, and the final volume \( V_2 \) is 250 mL. Your goal is to find \( M_2 \), the new molarity after dilution.Substitute these values into the dilution formula: \[4.0 \times 55 = M_2 \times 250\]Next, rearrange the formula to solve for \( M_2 \):\[M_2 = \frac{4.0 \times 55}{250}\]Carrying out the multiplication and division gives you:\[M_2 = \frac{220}{250} = 0.88 \ \text{M} \]So the new molarity of the diluted solution is 0.88 M. With these steps and formulas, you can confidently tackle any dilution problem by breaking it down into manageable parts.