Question

Consider a sugar solution (solution A) with concentration \(x .\) You pour one- third 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

The ratio of sugar in solutions A and B does not exist, as there is no sugar in solution B. The ratio of the volumes of solutions A and B is 2:1, with solution A having twice the volume of solution B. The ratio of the concentrations of sugar in solutions A and B does not exist, as there is no sugar concentration in solution B.

Step-by-step solution

Determine the ratio of sugar in solution A

Let the concentration of sugar in solution A be x. We pour one-third of this solution into a beaker, so the ratio of sugar in solution A now becomes 2/3x (as one-third is removed).

Determine the ratio of sugar in solution B

Since solution B is water, it has no sugar in it. So, the ratio of sugar in solution B is 0.

Compare the ratio of sugar in solutions A and B

The ratio of sugar in solutions A and B is: \( \dfrac{(\text {sugar in solution} \ A)}{(\text{sugar in solution} \ B)} \) Since there is no sugar in solution B, the denominator is zero, which makes the ratio undefined. Hence, the ratio of sugar in solution A and B does not exist. #b. Compare the volumes of solutions A and B.#

Determine the volume of solution B

To find the volume of solution B, recall that we're adding an equivalent volume of water to the one-third of solution A we removed. So, the volume of solution B is equal to one-third of the total volume of solution A.

Compare volumes of solutions A and B

Since one-third of solution A's volume was removed and the volume of the solution B is equal to the removed one-third of the solution A, the remaining volume of solution A is two-thirds of the original volume. Therefore, the volume of solution A is twice the volume of solution B. The ratio of the volumes of solutions A and B is 2:1. #c. What is the ratio of the concentrations of sugar in solutions A and B?#

Determine the concentration of sugar in solution A

Recall that the concentration of sugar in solution A is initially x. After removing one-third of the solution A, the concentration in the remaining two-thirds is still x.

Determine the concentration of sugar in solution B

The concentration of sugar in solution B is zero.

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

The ratio of the concentrations of sugar in solutions A and B is: \( \dfrac{(\text {concentration in solution} \ A)}{(\text{concentration in solution} \ B)} \) \( \dfrac{x}{0} \) Since there is no sugar in solution B, the denominator is zero and the ratio is undefined. Hence, the ratio of the concentrations of sugar in solutions A and B does not exist.

Key concepts

Dilution of Solutions

Understanding the process of dilution is fundamental in chemistry. Dilution involves adding a solvent, typically water, to a solution to decrease the concentration of solutes. In our exercise, solution A is diluted by adding an equivalent volume of water. Here are the essential points to grasp about dilution:

• When you dilute a solution, you are spreading out the solute particles by adding more solvent.
• The total amount of solute remains the same, but the volume of the solution increases, which lowers the solute concentration.
• The diluted solution (solution B) will have a lower concentration or no concentration of the solute, as it has been effectively 'watered down.'

In practical terms, if you have a sugar solution and add water to it, you won't have more sugar. Instead, the sugar that was in the original solution is now dispersed in a greater volume of liquid, making the solution less sweet, per unit of volume.

Volume and Concentration

The relationship between volume and concentration in a solution is an inverse one. When you increase the volume by adding solvent but keep the quantity of solute constant, as seen in our exercise with the sugar solution, the concentration decreases. This is because concentration is defined as the amount of solute per unit volume of solution. Here’s how this principle applied to our problem:

• Concentration is typically expressed in terms like molarity, which involves moles of solute per liter of solution.
• When solution A's volume is increased by adding water (creating solution B), the original concentration, represented as 'x', gets distributed across a larger volume.
• While the volume of solution A changes after dilution, the absolute amount of sugar remains unchanged, meaning the mass of solute is the same, but its concentration in solution B is zero since it’s pure water added.

In the context of our exercise, assuming solution A started with a specific volume, say 1L, and concentration of sugar 'x', then removing one-third volume and replacing it with water will still lead to a final volume of 1L, but the concentration of sugar will have dropped in the diluted portion because the same amount of sugar is now in a larger volume.

Ratio and Proportion in Chemistry

The concept of ratio and proportion plays a vital role in many chemical calculations, particularly when working with concentrations in solutions. A ratio compares two quantities, showing how much of one thing there is compared to another. In chemistry, it can refer to the ratio of solute to solvent, concentrations in different solutions, or the ratio of volumes. Proportion, on the other hand, is an equation that equates two ratios. In our exercise, the concept of ratio is used to compare quantities within the solutions:

• In the scenario with solution A and B, if we seek the ratio of the concentrations, we see that for solution B (pure water), the concentration of sugar is zero, making a ratio involving its concentration impossible to define.
• Using ratios allows chemists to scale up or down the volumes of solutions while maintaining the same concentration, which is handy for experiments requiring precise control over chemical conditions.

However, it's also important to understand that ratios involving a zero value in the denominator, as seen in the concentration comparison between solutions A and B, are undefined because division by zero has no meaning in mathematics and, by extension, in chemistry. This becomes particularly relevant when considering the practical applications, such as when making dilutions for experiments or industry processes.