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Chapter 4: Problem 3

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

Expert verified
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

01

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.
02

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.
03

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.
04

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.
05

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.

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