Question

You have a sugar solution (solution A) with concentration x. You pour one- fourth of this solution into a beaker, and add an equivalent volume of water (solution B). a. What is the ratio of sugar in solutions A and B? b. Compare the volumes of solutions A and B. c. What is the ratio of the concentrations of sugar in solutions A and B?

Short answer

a. The ratio of sugar in solutions A and B is 1:1. b. The volumes of solutions A and B have a ratio of 2:1, with solution A having twice the volume of solution B. c. The ratio of the concentrations of sugar in solutions A and B is 1:2, meaning solution A has twice the concentration of sugar compared to solution B.

Step-by-step solution

Calculate the volume of solution A

Initially, we have a sugar solution A with a concentration x. Let the total volume of solution A be V_A. When we pour one-fourth of solution A into the beaker to prepare solution B, the volume taken from solution A will be V_A/4.

Calculate the volume of solution B

To get solution B, we have poured V_A/4 of solution A into the beaker and added the same volume of water. So, the total volume of solution B will be the sum of the volume of solution A taken and the volume of added water, which is: V_A/4 + V_A/4 = V_A/2. Therefore, the volume of solution B is half of the volume of solution A.

Calculate the ratio of sugar in solutions A and B

Since we are adding water to solution A to get solution B, the total amount of sugar remains the same in both the solutions. Therefore, the ratio of sugar in solutions A and B will be 1:1 since the sugar amount is equal in both.

Compare the volumes of solutions A and B

We have already calculated the volume of solution B, which is half of the volume of solution A (V_A/2). Therefore, the volume ratio of solutions A and B can be given as V_A : V_A/2 or simply 2:1. This ratio tells us that solution A has twice the volume of solution B.

Calculate the ratio of the concentrations of sugar in solutions A and B

The concentration of solution A is x, which can be written as the mass of sugar in solution A (m_A) divided by the volume of solution A (V_A): \(C_A = \frac{m_A}{V_A}\). Since the amount of sugar remains the same in both solutions (m_A = m_B), we can write the concentration of solution B (C_B) as: \(C_B = \frac{m_A}{V_B}\), where \(V_B = V_A/2\). Now, let's find the ratio of the concentrations of sugar in solutions A and B: \( \frac{C_A}{C_B} = \frac{\frac{m_A}{V_A}}{\frac{m_A}{V_A/2}} = \frac{(m_A)(V_A/2)}{m_A V_A} = \frac{1}{2}\). So, the ratio of the concentrations of sugar in solutions A and B is 1:2. This tells us that solution A has twice the concentration of sugar compared to solution B.

Key concepts

Sugar Solution

When we talk about a sugar solution, we're referring to a mixture where sugar is dissolved in water. This is a common type of solution encountered in everyday life. The concentration of a sugar solution tells us how much sugar is present per unit of water. In our exercise, solution A is the initial sugar solution with a concentration denoted by \( x \).

Here's what's important to keep in mind about sugar solutions:

• Solvent and Solute: The sugar is the solute, and the water is the solvent.
• Concentration Measurement: The concentration can be given in various units, such as grams per liter or as a fraction like in our example.
• Homogeneous Mixture: This means the sugar is evenly distributed throughout the water.

A clear understanding of these points will help when adjusting or mixing solutions, as different processes can affect the concentration.

Volume Ratio

In our exercise, the volume ratio is crucial when comparing solutions A and B. Initially, the total volume of the sugar solution A is denoted as \( V_A \). To create solution B, one-fourth of this volume is taken, and an equivalent volume of water is added.

Here’s a breakdown of the volume ratio process:

• Step 1: Take one-fourth of solution A. This part is now \( \frac{V_A}{4} \).
• Step 2: Add water equal to this amount to create solution B. So, solution B ends up being \( \frac{V_A}{4} + \frac{V_A}{4} = \frac{V_A}{2} \).
• Understanding Ratios: The volume of solution A to B is 2:1. So, solution A has twice the volume of solution B.

This simple calculation helps us analyze how the volumes change and how much more or less one solution contains compared to another.

Concentration Ratio

The concentration ratio compares how much sugar per unit volume is in solutions A and B. Concentration is determined by the amount of solute divided by the volume of the solution. In our example, we start with solution A having a concentration \( x \).

Here's how we find the concentration ratio between solutions A and B:

• Concentration of A: \( C_A = \frac{m_A}{V_A} \), where \( m_A \) is the mass of sugar.
• Concentration of B: Since you pour a quarter from A into B and add equal water, \( C_B = \frac{m_A}{V_B} \) where \( V_B = \frac{V_A}{2} \).
• Ratio Formula: The concentration ratio \( \frac{C_A}{C_B} = \frac{(m_A)(V_A/2)}{m_A V_A} = \frac{1}{2} \).

This tells us solution A is twice as concentrated as solution B, since solution B's volume increases with the same sugar amount.

Dilution

Dilution is a process that involves decreasing the concentration of a solution by adding more solvent, which in this case is water. When solution A is diluted to create solution B, we increase the volume and thus decrease the sugar concentration.

Let's explore what happens during dilution:

• Initial Concentration: Solution A starts with concentration \( x \).
• Adding Water: By adding an equal volume of water to a portion of solution A, the total volume increases, causing the concentration of sugar to decrease.
• Final Concentration: Solution B's concentration reflects this dilution. It becomes half of what it was due to the volume doubling without changing the sugar mass.

Dilution is an essential concept in chemistry and practical applications, like changing the strength of flavored drinks or adjusting chemical reactions.