Ammonia Solution: Calculating Water Evaporation Equation

Have you ever wondered how to adjust the concentration of a solution by evaporating a solvent? It's a common practice in chemistry and various industries. Let's explore a specific scenario involving an ammonia solution and how we can determine the amount of water to evaporate to achieve a desired concentration. This article will provide a comprehensive explanation of how to set up the equation for this type of problem. We'll break down the concepts, walk through the steps, and make sure you understand the logic behind each part. So, whether you're a student tackling chemistry problems or simply curious about the science of solutions, you're in the right place!

Understanding the Problem: Ammonia Solution Concentration

In this problem, understanding the core concepts is key to formulating the correct equation. We start with 22 gallons of a 16% ammonia solution. This means that 16% of the total volume is ammonia, and the rest is water. Our goal is to evaporate some of the water to increase the ammonia concentration to 24%. The question asks us to find an equation that can be used to determine n, the number of gallons of water that need to be removed. To effectively solve this, we need to focus on the amount of ammonia, which remains constant throughout the evaporation process. The initial amount of ammonia can be calculated by multiplying the total volume of the solution by the initial concentration. This will serve as our baseline for comparison. We also need to consider how the total volume changes as water evaporates, which will directly impact the final concentration. The relationship between the amount of ammonia, the final volume, and the desired concentration is what we will express in our equation. By carefully setting up this equation, we can solve for n and find the amount of water needed to evaporate.

Initial Amount of Ammonia

The initial amount of ammonia is a crucial piece of information. We begin with 22 gallons of a 16% ammonia solution. To find the actual amount of ammonia, we multiply the total volume by the concentration. In this case, it's 22 gallons multiplied by 16%, or 0.16 as a decimal. This calculation gives us the number of gallons of pure ammonia present in the initial solution. This value is important because the amount of ammonia will not change when water is evaporated; only the concentration changes. This is because we're only removing water, not ammonia. Therefore, the initial amount of ammonia will be the same as the amount of ammonia in the final solution, which is a key concept in setting up our equation. Understanding this constant quantity allows us to relate the initial and final states of the solution through the amount of ammonia present. This forms the basis for creating an equation that accurately represents the evaporation process and helps us solve for the unknown variable, n, which represents the gallons of water evaporated.

Final Concentration

The final concentration, which is 24%, is the target we are trying to achieve by evaporating water. This means that after evaporating a certain amount of water, 24% of the remaining solution will be ammonia. The total amount of ammonia, which we calculated earlier, remains the same. However, the total volume of the solution decreases as water evaporates, leading to a higher concentration of ammonia. To express this mathematically, we consider the final volume of the solution, which will be the initial volume (22 gallons) minus the amount of water evaporated (n gallons). The final concentration is then the ratio of the amount of ammonia to the final volume. Setting up the equation involves equating the initial amount of ammonia to the final amount of ammonia, considering the change in volume. This relationship is what allows us to solve for n, the amount of water that needs to be evaporated to reach the 24% concentration. Understanding how the final concentration is related to the initial amount of ammonia and the change in volume is crucial for constructing the correct equation and solving the problem effectively.

Setting Up the Equation: Calculating the Unknown

To set up the equation, let's define our variables. Let n represent the number of gallons of water to be evaporated. The initial amount of ammonia in the solution is 22 gallons * 0.16 (16% expressed as a decimal), which equals 3.52 gallons. This value represents the pure ammonia content before any water is evaporated. The final volume of the solution after evaporating n gallons of water will be 22 - n gallons. The amount of ammonia remains constant at 3.52 gallons, but the concentration changes because the volume changes. The final concentration is given as 24%, or 0.24 as a decimal. Therefore, the final amount of ammonia can also be expressed as the final volume (22 - n) multiplied by the final concentration (0.24). This gives us the equation: 3. 52 = 0.24 * (22 - n). This equation represents the core relationship between the initial and final states of the solution, where the amount of ammonia remains constant. Solving this equation for n will give us the amount of water to be evaporated to achieve the desired 24% concentration.

Initial Ammonia Amount Equation

The initial ammonia amount can be calculated using a simple multiplication: the total initial volume of the solution multiplied by the initial concentration of ammonia. In our problem, we start with 22 gallons of a 16% ammonia solution. Converting the percentage to a decimal, 16% becomes 0.16. Therefore, the initial amount of ammonia is calculated as 22 gallons multiplied by 0.16, which equals 3.52 gallons. This calculation represents the actual volume of pure ammonia present in the solution before any water is evaporated. This value is a key constant in our equation because the amount of ammonia does not change during the evaporation process. Only the water content changes, which affects the overall concentration. By accurately calculating the initial amount of ammonia, we establish a solid foundation for comparing the initial and final states of the solution and for setting up an equation that correctly represents the relationship between these states. This step is crucial for finding the unknown variable, n, which represents the number of gallons of water to be evaporated.

Final Ammonia Amount Equation

The final ammonia amount needs to be expressed in terms of the final concentration and the final volume of the solution. We know that the amount of ammonia remains constant during the evaporation process, but the concentration changes because the volume of the solution decreases. The final concentration is given as 24%, which can be written as 0.24 in decimal form. The final volume is the initial volume minus the amount of water evaporated, which is 22 gallons - n gallons. Therefore, the final amount of ammonia can be expressed as 0.24 multiplied by (22 - n). This expression represents the amount of pure ammonia present in the solution after evaporating n gallons of water. By setting this expression equal to the initial amount of ammonia, we create an equation that reflects the conservation of ammonia during the evaporation process. This equation is the key to solving for n, the number of gallons of water that needs to be evaporated to achieve the desired 24% concentration. Understanding how to express the final amount of ammonia in terms of concentration and volume change is essential for accurately solving this type of problem.

Combining the Equations

Combining the equations is the critical step in solving for the unknown. We've established that the initial amount of ammonia is 3.52 gallons, and the final amount of ammonia can be expressed as 0.24 * (22 - n) gallons. Since the amount of ammonia remains constant during evaporation, we can set these two expressions equal to each other: 3.52 = 0.24 * (22 - n). This equation mathematically represents the relationship between the initial and final states of the ammonia solution. It states that the initial amount of pure ammonia is equal to the final concentration multiplied by the final volume. Solving this equation for n will give us the number of gallons of water that need to be evaporated to increase the concentration from 16% to 24%. By correctly combining the initial and final ammonia amount equations, we've created a powerful tool for solving the problem. This equation encapsulates the key principles of concentration and volume change and allows us to find the specific amount of water that needs to be removed to achieve the desired concentration.

The Correct Equation: Solution to the Puzzle

The correct equation that Grant can use to find n, the number of gallons of water he should remove, is A. rac{3.52}{22-n}=rac{24}{100}. Let's break down why this equation works and how it relates to our previous discussion. We know that the initial amount of ammonia is 3.52 gallons, and the final concentration is 24%. The denominator (22 - n) represents the final volume of the solution after evaporating n gallons of water. The left side of the equation, rac{3.52}{22-n}, represents the final concentration expressed as a decimal, which is the amount of ammonia divided by the final volume. The right side of the equation, rac{24}{100}, is the final concentration expressed as a fraction. By setting these two expressions equal to each other, we create an equation that directly relates the amount of ammonia, the change in volume, and the final concentration. This equation is the mathematical representation of the problem's core concept: the conservation of ammonia during evaporation. Solving this equation for n will give us the exact amount of water that needs to be removed to achieve the desired 24% ammonia solution.

Why Option A is Correct

Option A, rac{3.52}{22-n}=rac{24}{100}, is correct because it accurately represents the relationship between the initial and final states of the ammonia solution. The numerator on the left side, 3.52, is the initial amount of ammonia in gallons. The denominator, 22 - n, is the final volume of the solution after n gallons of water have been evaporated. The entire left side of the equation, therefore, represents the final concentration of ammonia in the solution, expressed as a ratio of ammonia volume to total volume. The right side of the equation, rac{24}{100}, is simply the desired final concentration, expressed as a fraction. By setting these two ratios equal to each other, we create an equation that allows us to solve for n, the unknown amount of water to be evaporated. This equation reflects the principle that the amount of ammonia remains constant during the evaporation process, while the concentration increases due to the reduction in volume. This makes option A the only choice that correctly captures the physical and chemical relationships at play in this problem.

In conclusion, understanding the concept of concentration, volume, and the conservation of solute is crucial for solving these types of problems. By breaking down the problem into smaller parts, we can formulate an equation that accurately represents the scenario and allows us to find the unknown variable. For further learning on solution concentrations and related topics, you can explore resources like Khan Academy's Chemistry Section.