Problem 115 - Introductory Chemistry Atoms First

Question

A solution of a particular solid solute in water has a concentration of \(25.0\) mass \(\%\). (a) Given \(100.0 \mathrm{~g}\) of this solution, how many grams of solute do you have? (b) Given \(48.0 \mathrm{~g}\) of this solution, how many grams of solute do you have? (c) How many grams of this solution do you need to obtain \(56.5 \mathrm{~g}\) of solute?

Step-by-Step Solution

Answer

(a) In 100.0 g of the solution, there are 25.0 g of solute. (b) In 48.0 g of the solution, there are 12.0 g of solute. (c) To obtain 56.5 g of the solute, you need 226.0 g of the solution.

Step 1: Write down the concentration formula for given situation

In this case, we have to find the mass of the solute in 100 g of the solution. To do this, we first write down the concentration formula as follows: Concentration (mass%) = (Mass of solute / Total mass) * 100

Step 2: Substitute the given values and solve for the mass of solute

Now, we have the concentration (25.0 mass%), and the total mass of the solution (100 g). Plugging these values into the formula: 25.0 = (Mass of solute / 100) * 100 To find the mass of solute, multiply both sides by 100: Mass of solute = 25.0 * 100 / 100

Step 3: Calculate the mass of solute

Now, we can calculate the mass of the solute by dividing both sides: Mass of solute = 25.0 g **Part (b)**

Step 1: Write down the concentration formula for given situation

In this case, we have to find the mass of the solute in 48.0 g of the solution. To do this, we first write down the concentration formula as follows: Concentration (mass%) = (Mass of solute / Total mass) * 100

Step 2: Substitute the given values and solve for the mass of solute

Now, we have the concentration (25.0 mass%), and the total mass of the solution (48.0 g). Plugging these values into the formula: 25.0 = (Mass of solute / 48.0) * 100 To find the mass of solute, multiply both sides by 48.0: Mass of solute = 25.0 * 48.0 / 100

Step 3: Calculate the mass of solute

Now, we can calculate the mass of the solute by dividing both sides: Mass of solute = 12.0 g **Part (c)**

Step 1: Write down the concentration formula for given situation

In this case, we have to find the mass of the solution needed to obtain 56.5 g of the solute. To do this, we first write down the concentration formula as follows: Concentration (mass%) = (Mass of solute / Total mass) * 100

Step 2: Substitute the given values and solve for the total mass of solution

Now, we have the concentration (25.0 mass%) and the mass of solute (56.5 g). Plugging these values into the formula: 25.0 = (56.5 / Total mass) * 100 To find the total mass of the solution, multiply both sides by the total mass and divide by 25.0: Total mass = 56.5 * 100 / 25.0

Step 3: Calculate the total mass of the solution

Now, we can calculate the total mass of the solution by dividing both sides: Total mass = 226.0 g So, to obtain 56.5 g of the solute, we need 226.0 g of the solution.

Key Concepts

Mass Percent

Mass percent, often abbreviated as mass \% or wt\%, is a way of expressing the concentration of a component in a mixture or solution. It represents the mass of an individual solute or component relative to the total mass of the solution or mixture, expressed as a percentage. This expression is widely used because of its simplicity and ease of interpretation.
To calculate mass percent, use the formula:

• \( \text{Mass \%} = \left( \frac{\text{Mass of solute}}{\text{Total mass of solution}} \right) \times 100 \)

Using this method, you can easily assess how much of a particular solute is in your solution relative to the total amount. Understanding mass percent is crucial for tasks such as preparing solutions with specific concentrations, comparing different solutions, or simply solving chemistry problems that involve concentration.

Solute Calculation

Calculating the mass of a solute in a solution is an important skill in chemistry, especially when working with concentrations expressed as mass percent. The process involves using the mass percent and total mass of the solution to find the mass of the solute.
Here are the steps performed:

• Use the given mass percent to find the mass of the solute, as per the formula \( \text{Mass of solute} = \left( \frac{\text{Mass percent}}{100} \right) \times \text{Total mass of solution} \)
• For example, with a 25.0 mass \% solution and a total solution mass of 100 g, the solute mass would be \( 25 = \left( \frac{25.0}{100} \right) \times 100 \)
• Simplifying, this calculates to 25 g of solute.

This approach can be adapted to find the solute mass in solutions with any given mass percent and total mass.

Mass of Solution

Determining the total mass of a solution when a specific mass of solute is desired requires an understanding of how mass percent works when rearranging the concentration formula.
To calculate the total mass of the solution needed to yield a desired mass of the solute:

• Use the derived formula \( \text{Total mass of solution} = \frac{\text{Mass of solute} \times 100}{\text{Mass percent}} \)
• This involves understanding how much of the solution is actually solute, as dictated by the mass percent.
• For instance, if 56.5 g of solute is required and the solution is 25.0 mass \%, substitute these values into the formula: \( 226 = \frac{56.5 \times 100}{25.0} \)
• The solution's total mass would be 226 g, having rearranged the concentration formula to solve for the total mass.

Understanding this calculation is essential when a precise amount of solute is needed for experiments or reactions.

Concentration Formula

The concentration formula is central to solving various problems in chemistry, particularly when specifying concentrations in terms of mass percent. It offers a straightforward relationship between solute mass, total solution mass, and concentration.
This formula is expressed as:

• \( \text{Concentration (mass\%)} = \left( \frac{\text{Mass of solute}}{\text{Total mass of solution}} \right) \times 100 \)

Using this, one can solve for any of these variables if the others are known. It is versatile and widely applicable in chemistry:

• To find the mass of solute when total mass and concentration are given, rearrange to \( \text{Mass of solute} = \left( \frac{\text{Mass percent} \times \text{Total mass of solution}}{100} \right) \)
• To determine the total mass of the solution needed to obtain a particular mass of solute, rearrange to \( \text{Total mass of solution} = \frac{\text{Mass of solute} \times 100}{\text{Mass percent}} \)

This fundamental formula is the foundation of many calculations, from analyzing mixture compositions to preparing solutions with precise concentrations.