Problem 52 - General Chemistry: Principles and Modern Applications

Question

A solution contains \(10.05 \%\) sucrose (cane sugar) by mass. What mass of the solution, in grams, is needed for an application that requires \(1.00 \mathrm{kg}\) sucrose?

Step-by-Step Solution

Answer

The mass of the solution required is 9950 grams.

Step 1 - Convert Percentage to Decimal

First, convert the percentage of sucrose to decimal by dividing the percentage by 100. So, 10.05% becomes 0.1005.

Step 2 - Identify the Given and Required Values

Identify and collect the data given in the problem. We need to find the total weight of the solution needed, and we know that the weight of sucrose required is 1 kg. Also, collect the percentage of sucrose in the solution in decimal form, which is 0.1005, obtained from step 1.

Step 3 - Calculation

Since we know that the sucrose is 0.1005 of the total solution mass, assign 0.1005x to the mass of sucrose, where x is the total mass of the solution. Since it's required that this mass of sucrose must be 1 kg, we write the equation 0.1005x = 1. Solve for x to obtain the total mass of the solution in kilogram.

Step 4 - Solution Conversion

The answer from the previous step is in kilograms. Convert it to grams (because the problem asks for the weight in grams) by multiplying by 1000 as 1 kilogram is equivalent to 1000 grams.

Key Concepts

Mass Percentage

Mass percentage is a way to express how much of one component is present in a solution compared to the total solution. It is calculated using the formula:\[\text{Mass Percentage} = \left( \frac{\text{Mass of Solute}}{\text{Total Mass of Solution}} \right) \times 100\]This formula helps us understand the proportion of the solute in the entire solution.
For example, if we have a sucrose solution with a mass percentage of 10.05%, it means that in every 100 grams of this solution, 10.05 grams are sucrose.
Knowing how to interpret this percentage is crucial for preparing solutions with accurate concentrations for various applications, whether it's in cooking, chemistry, or industrial processes.

• Mass percentage gives the ratio of solute mass to solution mass.
• It is useful for making solutions with precise concentration.
• Converting percentage to decimal format is often necessary for calculations.

Sucrose

Sucrose, commonly known as table sugar or cane sugar, is a type of carbohydrate that is sweet to taste and forms an essential ingredient in many foods. It is comprised of glucose and fructose molecules.
When working with solutions containing sucrose, understanding its concentration is important to ensure the desired sweetness or chemical reaction is achieved.
In the context of solutions, sucrose is considered as the solute, which is the component that is dissolved in the solvent to form a solution. For the exercise at hand, we are given a task where 10.05% of the solution's mass is consisted of sucrose.

• Sucrose = glucose + fructose.
• Used widely in culinary and industrial applications.
• In this context, it acts as the solute in the solution.

Understanding the role and proportion of sucrose in solutions can help in tasks such as sweetening products or preparing fermentation media.

Unit Conversion

Unit conversion is a fundamental skill in science and everyday problem-solving, allowing you to translate measurements from one unit to another. This is often necessary when dealing with problems where the units of the given quantities differ from the desired outcome.
In our exercise, we needed to convert kilograms of sucrose to grams because the problem asked for the solution mass in grams. Unit conversion is done using conversion factors, which are ratios that represent how one unit is related to another. For example:\[1 \text{ kg} = 1000 \text{ g}\]To convert from kilograms to grams, you can multiply the mass in kilograms by 1000.

• Understand the units you have and the units you want.
• Use conversion factors to transition between units.
• Multiply or divide as necessary based on the relationship between units.

Knowing how to convert between units ensures that you use the correct amounts in practical applications as well as scientific calculations.