Question

A weather balloon is filled with helium to a volume of \(1.61 \mathrm{~L}\) at 734 torr. What is the volume of the balloon after it has been released and its pressure has dropped to 0.844 atm? Assume that the temperature remains constant.

Short answer

The final volume of the balloon is 1.84 L.

Step-by-step solution

Convert pressures to a common unit

First, convert the given pressures to the same unit. The initial pressure is given as 734 torr, and the final pressure is 0.844 atm. Since 1 atm = 760 torr, we can convert the final pressure to torr: \[0.844 \text{ atm} \times \frac{760 \text{ torr}}{1 \text{ atm}} = 641.44 \text{ torr}\]

Use Boyle's Law

Since the temperature remains constant, use Boyle's Law, which states that for a given amount of gas, the pressure and volume are inversely related. The formula for Boyle's Law is: \[P_1 V_1 = P_2 V_2\]Given the initial volume (\(V_1 = 1.61 \text{ L}\)), initial pressure (\(P_1 = 734 \text{ torr}\)), and final pressure (\(P_2 = 641.44 \text{ torr}\)), solve for the final volume (\(V_2\))

Solve for the final volume

Rearrange the equation from Boyle's Law to solve for \(V_2\): \[V_2 = \frac{P_1 V_1}{P_2}\]Substitute the given values: \[V_2 = \frac{734 \text{ torr} \times 1.61 \text{ L}}{641.44 \text{ torr}} = 1.84 \text{ L}\]

Key concepts

Gas Laws

Gas laws are a set of rules that describe how gases behave. They help us understand the relationships between pressure, volume, and temperature for a given amount of gas. A key concept is that the behavior of gases can be predicted when one of these variables changes. There are several different gas laws, such as Boyle's Law, Charles's Law, and Avogadro's Law, each focusing on different aspects of gases. These laws are crucial in various fields, from weather forecasting to engineering.

Pressure-Volume Relationship

The pressure-volume relationship, often described by Boyle's Law, is one of the fundamental principles in the study of gases. According to Boyle's Law, the pressure of a given amount of gas is inversely related to its volume, as long as the temperature remains constant. This means that if you increase the pressure on a gas, its volume decreases, and if you decrease the pressure, its volume increases. Mathematically, Boyle's Law is stated as: \[P_1 V_1 = P_2 V_2\] where \(P_1\) and \(P_2\) are the initial and final pressures, while \(V_1\) and \(V_2\) are the initial and final volumes.

Constant Temperature

When studying gas behaviors, assuming constant temperature is often crucial. Under constant temperature conditions, the kinetic energy of the gas molecules doesn't change, which allows us to focus on how changes in pressure affect the volume and vice versa. For instance, in Boyle's Law problems such as the one with the weather balloon, the temperature remains unchanged. This stability helps us accurately apply the law without worrying about temperature effects. Constant temperature is vital for isolating and understanding pressure-volume changes in gases.