Problem 3 - Chemistry An Atoms First Approach

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 \(B ?\)

Step-by-Step Solution

Answer

In conclusion, a. The ratio of sugar in solutions A and B is 4:1. b. The ratio of the volumes of solutions A and B is 4:1. c. The ratio of the concentrations of sugar in solutions A and B is 1:1.

Step 1: Determine the volume and concentration of solution A

Let the volume of solution A be \(V_A\). Since we are taking one-fourth of this volume to create solution B, the volume of solution B is \(\frac{1}{4}V_A\). Adding an equivalent volume of water to this will not change the volume. Hence \(V_B = \frac{1}{4}V_A\). The concentration of solution A is given as \(x\).

Step 2: Calculate the mass of sugar in both solutions

We will now find the mass of sugar in both solutions. The mass of sugar in solution A, \(m_A\), can be calculated using the following formula: \[ m_A = xV_A \] Since we take one-fourth of solution A to create solution B, the mass of sugar in solution B, \(m_B\), will be one-fourth of the mass of sugar in solution A: \[ m_B = \frac{1}{4}m_A = \frac{1}{4}xV_A \]

Step 3: Determine the ratio of sugar in solutions A and B

The ratio of sugar between solutions A and B can be calculated using the ratio of their masses: \[ \frac{m_A}{m_B} = \frac{xV_A}{\frac{1}{4}xV_A} = 4 \] So, the ratio of sugar in solutions A and B is 4:1.

Step 4: Compare the volumes of solutions A and B

The volume of solution A is \(V_A\), and the volume of solution B is \(\frac{1}{4}V_A\). Since we are asked to compare the volumes of solutions A and B, we have: \[ \frac{V_A}{V_B} = \frac{V_A}{\frac{1}{4}V_A} = 4 \] So, the ratio of the volumes of solutions A and B is 4:1.

Step 5: Determine the ratio of the concentrations of sugar in solutions A and B

We can find the concentration of sugar in solution B by dividing the mass of sugar in solution B by its volume, \(V_B\): \[ concentration \: B = \frac{m_B}{V_B} = \frac{\frac{1}{4}xV_A}{\frac{1}{4}V_A} = x \] Since solution B is formed by diluting solution A with an equivalent volume of water, we can compare the concentrations of sugar in solutions A and B: \[ \frac{concentration \: A}{concentration \: B} = \frac{x}{x} = 1 \] So, the ratio of the concentrations of sugar in solutions A and B is 1:1. In conclusion, a. The ratio of sugar in solutions A and B is 4:1. b. The ratio of the volumes of solutions A and B is 4:1. c. The ratio of the concentrations of sugar in solutions A and B is 1:1.

Key Concepts

Sugar Solution

Understanding sugar solution is crucial in the context of dilution. Let's break it down easily. A sugar solution is essentially a mixture where sugar is the solute and water is the solvent. This mixture can vary in how much sugar it contains, which we refer to as its concentration. In this exercise, we're focusing on how to alter this concentration by adding more water—which is a common task in chemistry and everyday life. When we dilute a sugar solution, we decrease its concentration by adding more water. This means there's less sugar per unit of water. Imagine you're making lemonade: if you add more water to your mix, it tastes less sweet because the concentration of sugar is lower. This same principle applies with the sugar solution in the exercise. When one-fourth of the original sugar solution is poured out to form a new solution and then diluted with water, we effectively reduce the sugar content in the new solution, changing its concentration.

Concentration Ratio

The concentration ratio is a measure of how much sugar is present in each part of the solution. Think of it as a comparison of sweetness if you prefer. In our problem, we're looking at the concentration of sugar before and after dilution. In Solution A, the concentration is defined as the amount of sugar per volume, given by the formula: \[\text{Concentration} = \frac{\text{mass of sugar}}{\text{volume of solution}}\] Given that Solution B has the same mass of sugar but a larger volume due to the added water, the concentration decreases. The exercise asks us to compare these concentration ratios. Surprisingly, the calculation shows that the concentration of sugar in both solutions remains the same after diluting Solution B (1:1 ratio). This is because Solution B has the same proportional amount of sugar per volume as Solution A.

Volume Comparison

Volume comparison is a determining factor when we talk about dilution, impacting how concentrated a solution is. To understand this, picture the act of pouring one-fourth of a container of sugar water into a new container and adding an equivalent amount of water. Solution A originally had a volume denoted as \( V_A \). When forming Solution B, only one-fourth of Solution A is used, so its initial volume is \( \frac{1}{4} V_A \). After water is added in the same quantity, the volume of Solution B becomes \( \frac{1}{4} V_A + \frac{1}{4} V_A = \frac{1}{2} V_A \). Comparatively, Solution A with its full volume \( V_A \) is four times larger than the volume of diluted Solution B. Hence, the comparison of their volumes after dilution shows a ratio of 4:1 for Solution A to Solution B. This change highlights how dilution impacts not only concentration but also practical aspects like how much space the solution occupies.