Problem 55 - Precalculus with Limits

Question

BOYLE'S LAW: For a constant temperature, the pressure \(PL\) of a gas is inversely proportional to the volume \(V\) of the gas.

Step-by-Step Solution

Answer

Boyle's Law states that the product of pressure and volume is a constant for a given mass of confined gas as long as the temperature is unchanged. This is expressed by the formula \(P \cdot V = k\) or the formula \(P1 \cdot V1 = P2 \cdot V2\) when considering changes in pressure and volume.

Step 1: Understanding Boyle's Law

Boyle’s Law, named after Robert Boyle, is an experimental gas law that describes how the pressure of a gas tends to increase as the volume decreases when temperature remains constant. This law is represented as \(P \cdot V = k\), where \(P\) represents the pressure of the gas, \(V\) is the volume, and \(k\) is a constant.

Step 2: Applying Boyle's Law

We use Boyle's law to calculate the change in pressure or volume when one of these factors changes. For instance, if we know the initial pressure \(P1\) and volume \(V1\) and the volume changes to \(V2\), we can then find the new pressure \(P2\) by applying Boyle's Law. It can be expressed as \(P1 \cdot V1 = P2 \cdot V2\). Here, calculate \(P1 = \frac{P2 \cdot V2}{V1}\) if \(V2\), \(P2\), and \(V1\) are known and you need to find \(P1\). Similarly, calculate \(V1 = \frac{P2 \cdot V2}{P1}\) if \(V2\), \(P2\), and \(P1\) are known and you need to find \(V1\).

Step 3: Determining Values

Remember, in order to use Boyle's Law, we need to have at least three of the four variables. For instance, if you are given the initial pressure, initial volume, and the final pressure, you can easily find the final volume using Boyle's Law.

Key Concepts

Inverse Proportionality

Inverse proportionality is a fascinating mathematical concept that forms the cornerstone of Boyle's Law. When two quantities are inversely proportional, it means that as one quantity increases, the other decreases in a specific way. In simpler terms, if you have a set of seesaws, pushing down on one side causes the other side to rise.
This relationship is central to understanding Boyle's Law in gases:

• When the volume of a gas increases, the pressure decreases, assuming the temperature remains constant.
• Mathematically, this is expressed as \( P \cdot V = k \), where \( k \) is a constant.

As a result, when you double the volume of a gas, the pressure is halved, and vice versa. This inverse relationship is crucial for solving problems related to gas volumes and pressures.

Pressure and Volume Relationship

The pressure and volume relationship for gases is key to many practical applications, such as scuba diving, weather balloons, and even your car tires. According to Boyle's Law:

• Pressure \( P \) and volume \( V \) have an inverse relationship.
• The product \( P \cdot V \) remains constant at a fixed temperature.

This relationship implies that:

• When a gas is compressed, its pressure increases if the temperature doesn't change.
• A decreased gas volume means molecules are more crowded, leading to more collisions and higher pressure.

By understanding this relationship, you can predict how changes in the volume of a container will affect the gas within, under constant temperature conditions.

Gas Laws Calculation

Calculating changes in pressure or volume using gas laws requires a solid grasp of the formulas. Boyle's Law provides us with a simple formula to follow: \( P1 \cdot V1 = P2 \cdot V2 \).
This equation allows us to solve many practical problems:

• To find a new pressure \( P2 \), use: \( P2 = \frac{P1 \cdot V1}{V2} \).
• To find a new volume \( V2 \), use: \( V2 = \frac{P1 \cdot V1}{P2} \).

These calculations are essential, for instance, in determining how much gas a container can hold or predicting how changes in pressure will affect a gas's volume. Such calculations play a crucial role in chemistry and physics education, helping students apply theoretical concepts practically.

Experimental Gas Laws

Experimental gas laws, including Boyle's Law, are grounded in real-world observations and experiments. These laws provide formulas that describe how gases behave under various conditions:

• Boyle's Law deals with the inverse relationship between pressure and volume at constant temperature.
• This relationship is validated through experimentation, confirming that \( P \cdot V = k \) for a given mass of gas.

To experiment with Boyle's Law, one could vary the pressure of a gas using a piston to change its volume while maintaining constant temperature. Such experiments have wide-ranging implications in fields like engineering and meteorology.
By understanding these foundational principles and experimental behaviors, students can better appreciate the behavior of gases in everyday life.