Question

A gas has a volume of 450.0 mL. If the temperature is held constant, what volume would the gas occupy if the pressure were a. doubled? (Hint: Express \(P_{2}\) in terms of \(P_{r}, )\) b. reduced to one-fourth of its original value?

Short answer

When pressure is doubled, the volume is 225.0 mL. When pressure is reduced to one-fourth, the volume is 1800.0 mL.

Step-by-step solution

Understand Boyle's Law

Boyle's Law states that the pressure and volume of a gas are inversely proportional if the temperature is held constant: \[ P_1 V_1 = P_2 V_2 \]

Set Up the Equation for Part (a)

For part (a), the pressure is doubled, so we can express this as \( P_2 = 2P_1 \). Substitute this into Boyle's Law: \[ P_1 V_1 = 2P_1 V_2 \]

Solve for New Volume when Pressure is Doubled

Cancel \( P_1 \) from both sides of the equation: \[ V_1 = 2V_2 \] Substitute the known volume: \[ 450.0 = 2V_2 \] Solve for \( V_2 \): \[ V_2 = \frac{450.0}{2} = 225.0 \text{ mL} \]

Set Up the Equation for Part (b)

For part (b), the pressure is reduced to one-fourth, so \( P_2 = \frac{1}{4}P_1 \). Substitute this into Boyle's Law: \[ P_1 V_1 = \frac{1}{4}P_1 V_2 \]

Solve for New Volume when Pressure is One-Fourth

Cancel \( P_1 \) from both sides of the equation: \[ V_1 = \frac{1}{4}V_2 \] Substitute the known volume: \[ 450.0 = \frac{1}{4}V_2 \] Solve for \( V_2 \): \[ V_2 = 4 \times 450.0 = 1800.0 \text{ mL} \]

Key concepts

Gas Laws

The study of gas laws helps us understand how gases behave under different conditions of pressure, volume, and temperature. These laws are extremely useful in chemistry and physics.
One of the most fundamental gas laws is Boyle's Law, named after physicist Robert Boyle. Boyle's Law shows the relationship between the pressure and volume of a gas when the temperature is held constant. Other important laws include Charles's Law, which relates volume and temperature, and Avogadro’s Law, which connects volume and the number of gas molecules.
Understanding these laws can help predict how a gas will react if conditions like pressure or volume change.

Pressure-Volume Relationship

When exploring the behavior of gases, we often examine the pressure-volume relationship, which is precisely what Boyle's Law describes. According to Boyle's Law, the pressure and volume of a gas are inversely proportional as long as the temperature is constant.
In simpler terms, if you increase the pressure of a gas, its volume will decrease, assuming the temperature stays the same. Conversely, if you decrease the pressure, the volume will increase.
This relationship is mathematically represented by the equation \[ P_1 V_1 = P_2 V_2 \]
where:

• \(P_1\) and \(V_1\) are the initial pressure and volume
• \(P_2\) and \(V_2\) are the final pressure and volume

For example, if a gas initially has a volume of 450.0 mL and the pressure is doubled, using the formula you'll find that the new volume is 225.0 mL.

Inverse Proportionality

Inverse proportionality is a crucial concept in understanding Boyle's Law and the behavior of gases under different pressures. When two variables are inversely proportional, an increase in one results in a decrease in the other, and vice versa.
This is clearly demonstrated in Boyle's Law, where the product of the pressure and volume remains constant as long as the temperature does not change.
Let’s look at an example:

• If the pressure of a gas is quadrupled (that is reduced to one-fourth of its original pressure), the volume will increase by four times.

Using mathematical terms, if \[ P_2 = \frac{1}{4} P_1 \]then\[ V_2 = 4 V_1 \]
This means that for a gas with an initial volume of 450.0 mL, reducing the pressure to a quarter of its original value would result in a final volume of 1800.0 mL.
Understanding inverse proportionality is crucial for predicting how a gas will behave under different pressures and volumes.