Question

A sample of a gas at 0.780 atm occupies a volume of \(0.501 \mathrm{~L}\). If the temperature remains constant, what will be the new pressure if the volume increases to \(0.794 \mathrm{~L} ?\)

Short answer

The new pressure of the gas when the volume increases to \(0.794 \mathrm{~L}\) while maintaining a constant temperature is approximately \(0.491 \ \mathrm{atm}\).

Step-by-step solution

Understand Boyle's Law

Boyle's Law states that for a given mass of gas at constant temperature, the product of pressure and volume is constant. Mathematically, this can be represented as: \(P_1 V_1 = P_2 V_2\) where \(P_1\) and \(V_1\) are the initial pressure and volume, and \(P_2\) and \(V_2\) are the final pressure and volume.

List the given information

From the problem, we are given: Initial Pressure (\(P_1\)): 0.780 atm Initial Volume (\(V_1\)): 0.501 L Final Volume (\(V_2\)): 0.794 L We are tasked to find the new pressure (\(P_2\)).

Use Boyle's Law to find the new pressure

Using Boyle's Law equation, we can solve for the new pressure (\(P_2\)): \(P_1 V_1 = P_2 V_2\) Now plug in the given values: \(0.780 \ \mathrm{atm} \times 0.501 \ \mathrm{L} = P_2 \times 0.794 \ \mathrm{L}\)

Solve for P_2

To find the value of \(P_2\), divide both sides of the equation by the final volume (\(0.794 \ \mathrm{L}\)): \(P_2 = \frac{0.780 \ \mathrm{atm} \times 0.501 \ \mathrm{L}}{0.794 \ \mathrm{L}}\)

Calculate P_2

Now, perform the calculation: \(P_2 = \frac{0.780 \ \mathrm{atm} \times 0.501 \ \mathrm{L}}{0.794 \ \mathrm{L}} \approx 0.491 \ \mathrm{atm}\) The new pressure of the gas when the volume increases to \(0.794 \mathrm{~L}\) while maintaining a constant temperature is approximately \(0.491 \ \mathrm{atm}\).

Key concepts

Understanding Gas Laws

Gas laws describe how variables such as pressure, volume, and temperature interrelate and how they affect a sample of gas. These fundamental principles guide chemists and students to predict the behavior of gases under different conditions. Among these, Boyle's Law focuses solely on the pressure-volume relationship for a given mass of gas held at a constant temperature.

Gas laws prove essential in many real-life applications, from inflating balloons to managing the pressurized environment of a spacecraft. Mastering these laws allows us to solve a multitude of chemistry problems that involve gases in our surroundings.

Pressure-Volume Relationship in Boyle's Law

Boyle's Law provides a quantitative way to understand the inverse relationship between pressure and volume. It states that if the temperature of the gas remains constant, an increase in volume leads to a proportional decrease in pressure and vice versa. This relationship is critical when dealing with closed systems where temperature does not change.

To visualize this concept, imagine compressing a syringe: as you decrease the available volume for the air inside, the pressure increases because the air particles collide more frequently with the walls.

Tackling Chemistry Problems

When approaching chemistry problems, it's essential to break them down step by step, as with our original exercise. Identification of the known variables and the target variable kickstarts the problem-solving process. Subsequently, a relevant chemical law is applied—in our case, Boyle's Law—to establish the relationship between the quantities.

Chemistry often involves converting and managing different units of measurement, so pay careful attention to ensure accuracy. Use appropriate formulas, and don't hesitate to recheck calculations to prevent errors.

Problem-Solving in Chemistry

Effective problem-solving in chemistry calls for a solid understanding of the concepts at play and the ability to apply them to new situations. In our exercise, Boyle's Law was used to identify the final pressure of a gas when its volume changed. After setting up the appropriate equation, the problem was solved algebraically. This step-by-step method can be applied to various other chemistry problems, providing a systematic approach to obtaining solutions.

Remember to always review the units and ensure that they are compatible with each other. This type of methodical approach fosters a deeper comprehension and enables students to tackle complex problems with confidence.