Question

A sample of nitrogen \(\left(\mathrm{N}_{2}\right)\) has a volume of \(50.0 \mathrm{~L}\) at a pressure of \(760 . \mathrm{mmHg}\). What is the volume, in liters, of the gas at each of the following pressures, if there is no change in temperature and amount of gas? a. \(1500 \mathrm{mmHg}\) b. \(4.00 \mathrm{~atm}\) c. \(0.500 \mathrm{~atm}\)

Short answer

a) 25.3 L; b) 12.5 L; c) 100 L

Step-by-step solution

- Understand Boyle's Law

Boyle's Law states that the volume of a gas is inversely proportional to its pressure when the temperature and quantity of gas remain constant. This can be expressed as: \[ P_1 V_1 = P_2 V_2 \]

- Identify given values

The initial volume (\(V_1\)) is \(50.0\) L. The initial pressure (\(P_1\)) is \(760\) mmHg.

- Solve for new volume at \(1500 \,\text{mmHg}\)

Convert pressures into same unit if needed:\(P_2 = 1500\) mmHg.Using Boyle’s law \( P_1 V_1 = P_2 V_2\): \[ 760 \, mmHg \times 50.0 \, L = 1500 \, mmHg \times V_2 \]Solve for \(V_2\): \[ V_2 = \frac{760 \, mmHg \times 50.0 \, L}{1500 \, mmHg} \] \[ V_2 \approx 25.3 \, L \]

- Solve for new volume at \(4.00 \,\text{atm}\)

Convert \(4.00\) atm to mmHg since \(1\) atm = \(760\) mmHg. Therefore, \(P_2 = 4.00\) atm \(× 760\) mmHg/atm = \(3040\) mmHg.Using Boyle’s law \( P_1 V_1 = P_2 V_2 \): \[ 760 \, mmHg \times 50.0 \, L = 3040 \, mmHg \times V_2 \]Solve for \(V_2\): \[ V_2 = \frac{760 \, mmHg \times 50.0 \, L}{3040 \, mmHg} \] \[ V_2 \approx 12.5 \, L \]

- Solve for new volume at \(0.500 \,\text{atm}\)

Convert \(0.500\) atm to mmHg since \(1\) atm = \(760\) mmHg. Therefore, \(P_2 = 0.500\) atm \(× 760\) mmHg/atm = \(380\) mmHg.Using Boyle’s law \( P_1 V_1 = P_2 V_2 \): \[ 760 \, mmHg \times 50.0 \, L = 380 \, mmHg \times V_2 \]Solve for \(V_2\): \[ V_2 = \frac{760 \, mmHg \times 50.0 \, L}{380 \, mmHg} \] \[ V_2 = 100 \, L \]

Key concepts

Gas Laws

Gas laws help us understand how gases behave under different conditions. Boyle's Law is one such principle. These laws are essential for many fields of science and engineering. Gas laws describe the relationships between volume, temperature, pressure, and quantity of a gas.
They provide the theoretical basis for understanding real-world phenomena. For instance, when you breathe, gas laws help explain how air flows in and out of your lungs.
Boyle's Law is particularly focused on the relationship between pressure and volume, simplifying the calculations when temperature and the amount of gas remain constant.

Pressure-Volume Relationship

Boyle's Law specifically addresses the pressure-volume relationship of a gas. It states that the volume of a gas is inversely proportional to its pressure. This means that if the pressure increases, the volume decreases, and vice versa. The formula is represented as: \[ P_1 V_1 = P_2 V_2 \] Where \(P_1\) is the initial pressure, \(V_1\) is the initial volume, \(P_2\) is the final pressure, and \(V_2\) is the final volume. This equation allows us to calculate the new volume when the gas pressure changes. Understanding this relationship helps in various applications such as in syringes, hydraulic systems, and understanding atmospheric behaviors. To solve any problem involving Boyle's Law, you just need two sets of known pressure and volume values.

Inverse Proportionality

Inverse proportionality is a core concept to grasp in Boyle’s Law. When two variables are inversely proportional, as one variable increases, the other decreases proportionally. For Boyle's Law, this means that if you double the pressure exerted on a gas, its volume will be halved given constant temperature and amount of gas. This relationship can be visualized through experiments and practical examples. For example, if you squeeze a balloon (increasing the pressure), the balloon's volume decreases. Understanding inverse proportionality helps to make predictions about how systems behave. In Boyle's Law, it lets us predict how a gas will respond to changes in pressure, making this concept very useful in daily life and scientific research.