Problem 20 - Chemistry

Question

A gas occupying a volume of \(725 \mathrm{~mL}\) at a pressure of 0.970 atm is allowed to expand at constant temperature until its pressure reaches 0.541 atm. What is its final volume?

Step-by-Step Solution

Answer

The final volume of the gas is approximately 1305 mL.

Step 1: Identify given variables

The initial pressure, P1 is 0.970 atm. The initial volume, V1 is 725 mL. The final pressure, P2 is 0.541 atm.

Step 2: Apply Boyle's Law

We should apply the formula of Boyle's law, \(P1 × V1 = P2 × V2\). We can rearrange this formula to solve for the final volume \(V2\), that gives us \(V2 = (P1 × V1) / P2\).

Step 3: Substitute the values

By substituting the values into the formula \(V2 = (0.970 × 725) / 0.541\), we will be able to find the final volume.

Step 4: Get the final volume

Performing the calculation gives us a value of approximately 1305 mL for the final volume.

Key Concepts

Gas Laws

Gas laws are essential in understanding how gases behave under different conditions. These laws provide formulas that relate the pressure, volume, and temperature of a gas.
Two of the most commonly used gas laws are Boyle's Law and Charles's Law. Boyle's Law connects pressure and volume at constant temperature, while Charles's Law connects volume and temperature at constant pressure.

• Boyle's Law: This law states that for a fixed amount of gas at a constant temperature, the volume is inversely proportional to the pressure.
• Charles's Law: This states that for a fixed amount of gas at constant pressure, the volume is directly proportional to the temperature in Kelvin.

Understanding these laws helps predict how a gas will behave when its conditions change, such as in weather balloons or car tires.

Volume and Pressure Relationship

The relationship between volume and pressure in a gas is key to many applications.
Boyle's Law, which we used in the exercise, explains this concept perfectly. It states that for a constant temperature, the product of the pressure and volume of a gas is constant.
This can be mathematically represented as:
\[ P_1 \times V_1 = P_2 \times V_2 \]
Rearranging this equation gives us the formula to calculate the unknown volume or pressure.

• When the pressure of a gas increases, its volume reduces if the temperature remains unchanged. This is because there's less space left for gas molecules to move around.
• Conversely, when the pressure decreases, the gas molecules can spread out more, and hence, the volume increases.

This principle is used in scuba diving, where the varying pressure affects the air inside the diver's tank.

Ideal Gas Behavior

The concept of ideal gas behavior is a simplified way to understand how gases behave under various conditions. In reality, gases don't always behave perfectly, but we use this model for easier calculations.
An ideal gas is assumed to have molecules that occupy no volume and experience no intermolecular forces.

• Real gases deviate from ideal behavior at high pressures and low temperatures.
• The Ideal Gas Law combines Boyle's, Charles's, and Avogadro's laws into one formula: \[ PV = nRT \]
• Here, \(P \) is pressure, \(V \) is volume, \(n \) is the number of moles, \(R \) is the gas constant, and \(T \) is the temperature in Kelvin.

Using the ideal gas model helps with calculations involving a mixture of gases, like in chemical reactions, even when accounting for real gas behavior might be too complicated.