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Chapter 12: Problem 56

How many grams of a \(\mathrm{CaCl}_{2}(a q)\) solution that is \(14 \% \mathrm{CaCl}_{2}\) by mass contains \(3.25\) grams of \(\mathrm{CaCl}_{2}(s) ?\)

Short Answer

Expert verified
The solution weighs approximately 23.21 grams.

Step by step solution

01

Understand the Problem

We need to find the total mass of a calcium chloride solution where 14% of the solution's mass is composed of calcium chloride (\( \mathrm{CaCl}_{2} \)). We are given that this solution should contain 3.25 grams of calcium chloride.
02

Set Up the Percentage Equation

We know that to find the total mass of the solution, we use the equation for mass percentage: \[ \text{mass %} = \left(\frac{\text{mass of solute}}{\text{total mass of solution}}\right) \times 100 \]. In this scenario, the solute is calcium chloride.
03

Substitute Known Values

Plug the given values into the percentage equation: \( 14 = \left(\frac{3.25}{x}\right) \times 100 \), where \( x \) is the total mass of the solution in grams.
04

Solve for the Total Mass

To find \( x \), the total mass of the solution, rearrange the equation and solve: \[ x = \frac{3.25 \times 100}{14} \]. Evaluate this to find \( x \).
05

Calculate the Result

Perform the calculation: \[ x = \frac{325}{14} \approx 23.21 \]. Therefore, the total mass of the solution is approximately 23.21 grams.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Mass Percentage
Mass percentage is a simple yet powerful concept in chemistry. It tells you how much of a solute is present in a solution, expressed as a fraction of the total mass of the solution. The formula for mass percentage is:
  • mass % = \( \left( \frac{\text{mass of solute}}{\text{total mass of solution}} \right) \times 100 \)
This formula helps chemists and students understand the composition of a solution without needing the quantities of all components.
In our exercise, we know the mass percentage of calcium chloride (\(\mathrm{CaCl}_2\)) is 14%. This means that in every 100 grams of solution, 14 grams is calcium chloride. This simplification allows for quick calculations and a better understanding of solution compositions.
Mass percentage is crucial in lab work and industry, ensuring the precise preparation of solutions for reactions and processes.
What Is Calcium Chloride?
Calcium chloride, often found in the compound form as \(\mathrm{CaCl}_2\), is a salt that is commonly used in various applications. It's used to de-ice roads in winter by lowering the freezing point of water.
You might also find it in food processing, as it can help in firming canned vegetables and as an electrolyte in sports drinks.
Understanding how calcium chloride behaves when dissolved in water or when used in mixtures is important. It is highly soluble, meaning it dissolves readily in water, making it an excellent choice for applications where rapid dissolution is needed.
Solute and Solvent Relationship
The solute and solvent relationship is foundational in chemistry. A solution is a mixture of solute, which is the substance being dissolved, and solvent, the substance doing the dissolving.
  • In this exercise, \(\mathrm{CaCl}_2\) is the solute, and the water or aqueous solution is the solvent.
The ability of a solute to dissolve in a solvent depends on several factors including temperature, pressure, and the nature of the solute and solvent themselves.
Solutions can be either saturated, unsaturated, or supersaturated depending on how much solute is dissolved.
The equation used to find the mass percentage relates these components directly, showing how much of the solute is present in the total mass of the solution. This is crucial for calculating how much of both components is needed to create solutions with specific properties.

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