Question

The volume of air in a person's lungs is \(615 \mathrm{~mL}\) at a pressure of \(760 . \mathrm{mmHg} .\) Inhalation occurs as the pressure in the lungs drops to \(752 \mathrm{mmHg}\) with no change in temperature and amount of gas. To what volume, in milliliters, did the lungs expand?

Short answer

The lungs expanded to approximately 621.8 mL.

Step-by-step solution

Understand Boyle's Law

Boyle's Law states that for a given mass of gas at constant temperature, the pressure and volume are inversely proportional. Mathematically, it is represented as \(P_1 V_1 = P_2 V_2\). Where \(P_1\) and \(P_2\) are the initial and final pressures, and \(V_1\) and \(V_2\) are the initial and final volumes respectively.

Identify Given Values

Use the values given in the problem statement: \(P_1 = 760 \, \mathrm{mmHg}\), \(V_1 = 615 \, \mathrm{mL}\), and \(P_2 = 752 \, \mathrm{mmHg}\).

Set Up Boyle's Law Equation

Substitute the given values into the equation \(P_1 V_1 = P_2 V_2\): \[760 \, \mathrm{mmHg} \times 615 \, \mathrm{mL} = 752 \, \mathrm{mmHg} \times V_2\].

Solve for \( V_2 \)

Rearrange the equation to isolate \(V_2\): \[V_2 = \frac{760 \, \mathrm{mmHg} \times 615 \, \mathrm{mL}}{752 \, \mathrm{mmHg}}\].

Calculate the Final Volume

Perform the calculation: \[V_2 = \frac{760 \, \mathrm{mmHg} \times 615 \, \mathrm{mL}}{752 \, \mathrm{mmHg}} \approx 621.8 \, \mathrm{mL}\].

Key concepts

gas laws

Gas laws help us understand how gases behave under different conditions. They describe the relationship between pressure, volume, temperature, and the amount of gas. These laws include Boyle's Law, Charles's Law, and the Ideal Gas Law. Each of these laws holds true under certain conditions, such as constant temperature or pressure. Boyle's Law focuses on the pressure-volume relationship, which is crucial for understanding how gases expand or compress. Learning these laws will give you a solid foundation to solve problems involving gases.

pressure-volume relationship

The pressure-volume relationship, explained by Boyle's Law, states that for a fixed amount of gas at constant temperature, the pressure and volume are inversely proportional. This means if one increases, the other must decrease. Mathematically, it is represented as:
\(\)
In simpler terms, if you compress a gas into a smaller volume, its pressure goes up, and vice versa. This principle is vital for various applications, from understanding breathing to the working of pistons and syringes. The pressure-volume relationship helps us predict how changes in one variable affect the other.

inverse proportionality

Inverse proportionality means that as one value increases, the other decreases. In the context of Boyle's Law, pressure and volume are inversely proportional. If you imagine inflating a balloon, as you blow more air into it (increasing volume), the pressure inside the balloon rises until it balances with the pressure outside. Similarly, when you inhale, the pressure in your lungs drops, causing volume to increase. Understanding inverse proportionality helps us explain many natural phenomena and engineering designs involving gases. It shows how interconnected variables affect one another.