Question

A gas occupying a volume of \(725 \mathrm{~mL}\) at a pressure of 0.970 atm is allowed to expand at constant temperature until its pressure reaches \(0.541 \mathrm{~atm} .\) What is its final volume?

Short answer

The final volume of the gas is 1370 mL.

Step-by-step solution

Identify Given Values

Identify the given values from the problem. The initial volume \(V_1 = 725 mL\), the initial pressure \(P_1 = 0.970 atm\), and the final pressure \(P_2 = 0.541 atm\). We are asked to find the final volume \(V_2\).

Apply Boyle's Law

Based on Boyle's Law, \(P_1V_1 = P_2V_2\). Rearrange the equation to solve for \(V_2\), the unknown, which results in \(V_2= \frac{P_1V_1}{P_2}\).

Perform Calculations

Substitute the known values into the equation from step 2: \(V_2 = \frac{(0.970 atm) * (725 mL)}{0.541 atm}\). When you perform the calculation \(V_2 = 1370 mL\).

Key concepts

Gas Laws Calculations

Gas laws are foundational concepts in chemistry that describe the behavior of gases under various conditions of pressure, volume, and temperature. To calculate changes in gas properties, you need to have a good grasp of these laws and how they interact.

One of the primary principles in gas laws calculations is understanding the relationship between pressure and volume under a constant temperature, famously known as Boyle's Law. Other important relationships are described by Charles's Law, which relates volume and temperature, and Gay-Lussac's Law, which correlates pressure and temperature. Together, these principles are combined into the Ideal Gas Law, which provides a comprehensive equation to solve for any one of the variables, given that the others are known.

To tackle calculations in this area, it's crucial to:

Identify the known variables and what you need to find,

Determine which gas law applies to the given conditions,

Rearrange the equation to solve for the unknown variable, and

Pay attention to units and convert them if necessary to ensure consistency.

Remember, maintaining the correct units throughout your calculation is essential for your results to be accurate.

Boyle's Law Equation

Boyle's Law focuses on the inverse relationship between the pressure and volume of a confined gas at constant temperature. Its equation is elegantly simple: \( P_1V_1 = P_2V_2 \).

To use Boyle's Law, the equation can be reshaped depending on what you're solving for. If you're finding volume, the equation becomes \( V_2 = \frac{P_1V_1}{P_2} \) as we saw in the textbook example. If it’s the pressure you're after, flip it around to \( P_2 = \frac{P_1V_1}{V_2} \).

Remember these important notes when applying Boyle's Law:

The temperatures must remain constant throughout the experiment,

Pressure and volume are inversely proportional, so if one increases, the other decreases, and vice versa,

Always use consistent units for pressure and volume.

If sometimes units are given in different systems (like mL for volume and atmospheres for pressure), no conversion is needed as long as they are consistently used on both sides of the equation. Moreover, understanding the real-life applications of Boyle's Law, such as in human breathing or syringes, can deepen comprehension.

Chemistry Problem Solving

Solving problems in chemistry can seem daunting, but it becomes manageable by following a structured approach. Here’s how to enhance your problem-solving skills for chemistry:

First, carefully read and understand the problem. What are you asked to find, and what information has already been given? Make a list of known quantities and denote the unknowns clearly. Secondly, visualize the problem if possible. Creating a diagram can help you see the relationships between variables.

Thirdly, pick the correct equation that applies to your problem, such as Boyle’s Law for pressure-volume relationships at constant temperature. Ensure you understand how to manipulate the equation to isolate your unknown variable. Next, substitute the known values into your rearranged equation and solve for the unknown. Here, it’s vital to pay attention to the units and convert them if necessary to ensure consistency and accuracy.

In chemistry, it's also imperative to keep significant figures in mind, as precision matters in scientific calculations. Lastly, after solving the problem, reassess to ensure that the answer is plausible based on the context of the question. Does it make physical sense? If not, you may need to revisit your steps and look for any errors.