Helium Gas Expansion Calculation Initial Quantity
In the realm of chemistry, understanding the behavior of gases is fundamental, particularly when dealing with changes in volume, pressure, temperature, and the amount of gas present. This article delves into a classic scenario involving the expansion of a balloon filled with helium gas. We will explore how the principles of gas laws can be applied to determine the initial quantity of helium in the balloon, given its volume change and the final amount of gas. This is a practical application of Avogadro's Law, which states that the volume of a gas is directly proportional to the number of moles when temperature and pressure are kept constant. Grasping this concept is crucial not only for chemistry students but also for anyone interested in the properties of matter and the relationships between different variables in a gaseous system.
The problem at hand presents a balloon initially containing a certain amount of helium gas. As more helium is added, the balloon expands, indicating a direct relationship between the amount of gas and the volume it occupies. The initial volume of the balloon is given as 230 mL, and it expands to a final volume of 860 mL after more helium is added. The final amount of helium in the expanded balloon is quantified as 3.8 x 10^-4 moles. Our task is to determine the initial amount of helium present in the balloon before the addition, assuming that the temperature and pressure remain constant throughout the process. This scenario exemplifies a real-world application of gas laws, where we can calculate the quantity of a gas based on its volume changes under controlled conditions.
To effectively solve this problem, we will employ Avogadro's Law, a cornerstone principle in the study of gases. Avogadro's Law provides a direct correlation between the volume of a gas and the number of moles it contains, given constant temperature and pressure. This means that as the number of moles of gas increases, the volume also increases proportionally, and vice versa. By understanding and applying this law, we can establish a relationship between the initial and final states of the helium gas in the balloon. We will set up a proportion that relates the initial volume and number of moles to the final volume and number of moles. This will allow us to isolate and calculate the unknown initial quantity of helium. This step-by-step approach will not only provide the numerical answer but also reinforce the understanding of how gas laws govern the behavior of gases in various situations.
Let's clearly define the helium expansion scenario. This involves understanding the given parameters and what we are trying to find. We have a balloon that initially contains helium gas, with an initial volume of 230 mL. More helium is then added to the balloon, causing it to expand. The final volume of the balloon after the addition is 860 mL. We are also given that the expanded balloon contains 3.8 x 10^-4 moles of helium. The key assumption here is that the temperature and pressure remain constant throughout this process. This is crucial because it allows us to apply Avogadro's Law, which is valid only under constant temperature and pressure conditions. Our primary objective is to determine the initial quantity of helium present in the balloon before any additional gas was introduced. This means we need to find the initial number of moles of helium.
To approach this problem systematically, let's identify the knowns and the unknown. The knowns are the initial volume (V1), the final volume (V2), and the final number of moles (n2). The unknown is the initial number of moles (n1). By clearly defining these variables, we set the stage for applying the appropriate gas law to solve for the unknown. This step is essential in any quantitative problem-solving process, as it helps to organize the information and identify the relevant relationships between variables. Carefully noting the knowns and unknowns prevents confusion and ensures that we are targeting the correct quantity in our calculations.
Furthermore, it's important to recognize the underlying principle that governs this scenario: Avogadro's Law. Avogadro's Law states that equal volumes of all gases, at the same temperature and pressure, contain the same number of molecules. This law implies a direct proportionality between volume and the number of moles of gas. Mathematically, this relationship can be expressed as V ∝ n, where V represents volume and n represents the number of moles. This proportionality constant allows us to set up a ratio between the initial and final states of the gas. Understanding this fundamental relationship is key to solving the problem correctly. In the next section, we will delve into the application of Avogadro's Law to this specific problem, setting up the necessary equations and performing the calculations to find the initial quantity of helium.
To solve for the initial quantity of helium, we will use Avogadro's Law, which, as mentioned earlier, states that the volume of a gas is directly proportional to the number of moles when temperature and pressure are constant. This relationship can be expressed mathematically as V ∝ n, where V is the volume and n is the number of moles. To apply this law to our problem, we can set up a proportion that relates the initial and final states of the gas. This is done by creating a ratio of the initial volume (V1) to the initial number of moles (n1) and equating it to the ratio of the final volume (V2) to the final number of moles (n2). The proportion can be written as: V1 / n1 = V2 / n2
This equation is the cornerstone of our solution. It allows us to relate the given volumes and final number of moles to the unknown initial number of moles. The equation is derived directly from Avogadro's Law and highlights the direct proportionality between volume and the number of moles. By setting up this proportion, we have effectively translated the conceptual understanding of Avogadro's Law into a mathematical framework that we can use to solve for the unknown. The next step involves plugging in the known values into this equation and rearranging it to isolate and solve for n1, the initial number of moles of helium.
Before we proceed with the calculations, it's essential to ensure that the units are consistent. In this case, the volumes are given in milliliters (mL), and the number of moles is given in moles (mol). Since we are setting up a proportion, the units for volume will cancel out, so we don't need to convert mL to liters. However, it's always a good practice to double-check the units to avoid any errors in the final result. The key here is consistency; as long as the volumes are in the same unit on both sides of the equation, the calculation will be accurate. Now, with the proportion set up and the units verified, we are ready to substitute the given values and solve for the initial number of moles of helium. This process will demonstrate the practical application of Avogadro's Law in a real-world scenario.
With the proportion V1 / n1 = V2 / n2 established, we can now plug in the known values to solve for the initial number of moles (n1). We are given: Initial volume, V1 = 230 mL, Final volume, V2 = 860 mL, Final number of moles, n2 = 3.8 x 10^-4 mol. Substituting these values into the proportion, we get: 230 mL / n1 = 860 mL / (3.8 x 10^-4 mol). To solve for n1, we can cross-multiply and rearrange the equation. Cross-multiplying gives us: 230 mL * (3.8 x 10^-4 mol) = 860 mL * n1. Now, we can isolate n1 by dividing both sides of the equation by 860 mL: n1 = (230 mL * 3.8 x 10^-4 mol) / 860 mL. This rearrangement is a crucial step in isolating the unknown variable and setting up the equation for the final calculation.
Performing the calculation, we get: n1 = (230 * 3.8 x 10^-4) / 860 mol. First, multiply 230 by 3.8 x 10^-4: 230 * 3.8 x 10^-4 = 0.0874. Now, divide this result by 860: 0.0874 / 860 ≈ 1.016 x 10^-4. Therefore, the initial number of moles of helium, n1, is approximately 1.016 x 10^-4 mol. This numerical result represents the quantity of helium gas initially present in the balloon before the addition of more helium. It's important to note that this calculation is based on the assumption that the temperature and pressure remain constant, which allows us to apply Avogadro's Law.
To summarize the calculation, we used the given volumes and the final number of moles to set up a proportion based on Avogadro's Law. We then cross-multiplied and rearranged the equation to solve for the initial number of moles. The final result, approximately 1.016 x 10^-4 mol, provides a quantitative answer to the problem statement. This step-by-step calculation not only provides the solution but also reinforces the understanding of how to apply gas laws in practical scenarios. In the next section, we will discuss the significance of this result and its implications in the context of gas behavior and Avogadro's Law.
In conclusion, by applying Avogadro's Law, we determined that the initial quantity of helium present in the balloon was approximately 1.016 x 10^-4 moles. This result is significant because it demonstrates the direct relationship between the volume of a gas and the number of moles it contains, given constant temperature and pressure. The problem presented a scenario where a balloon expanded as more helium was added, and by using the principles of gas laws, we were able to quantitatively determine the initial amount of gas. This exercise highlights the practical application of Avogadro's Law in understanding and predicting the behavior of gases.
The result also underscores the importance of Avogadro's Law in the field of chemistry. Avogadro's Law is a fundamental principle that helps us relate macroscopic properties of gases, such as volume, to microscopic quantities, such as the number of moles. This connection is crucial for many chemical calculations and applications, including stoichiometry, gas reactions, and the determination of molar masses. Understanding Avogadro's Law allows chemists to make accurate predictions and calculations about gas behavior, which is essential in various scientific and industrial processes.
Furthermore, this problem-solving exercise reinforces the importance of a systematic approach to quantitative problems in chemistry. By clearly defining the problem, identifying the knowns and unknowns, setting up the appropriate equations, and performing the calculations carefully, we were able to arrive at a meaningful solution. This methodical approach is not only applicable to gas law problems but also to a wide range of chemical calculations. It emphasizes the importance of understanding the underlying principles, setting up the problem correctly, and executing the calculations with precision. The ability to apply these skills is crucial for success in chemistry and related fields. Therefore, the solution to this problem not only provides a numerical answer but also reinforces key concepts and problem-solving strategies in chemistry.