Question

Show how Charles's gas law can be derived from the ideal gas law.

Short answer

To derive Charles's gas law from the ideal gas law, begin with the ideal gas law formula: \(PV = nRT\). Isolate the volume (V) by dividing both sides by pressure (P) and the ideal gas constant (R), then multiplying by the amount of gas (n), resulting in: \(V = \frac{nRT}{P}\). Assume constant pressure and amount of gas, treating n, R, and P as constants in the equation, giving: \(V = kT\), where k is a constant representing \(\frac{nR}{P}\). This relationship, \(V = kT\), is Charles's Gas Law, showing that the volume of a gas is directly proportional to its temperature at constant pressure.

Step-by-step solution

Identify the Ideal Gas Law

The ideal gas law can be given by: \(PV = nRT\) Where: \(P\) = Pressure of the gas \(V\) = Volume of the gas \(n\) = Amount of the gas in moles \(R\) = Ideal gas constant \(T\) = Temperature of the gas in Kelvin

Isolate Volume in the Ideal Gas Law Formula

In order to derive Charles's gas law, we need to make Volume (V) the subject of the formula. This can be done by dividing both sides of the equation by Pressure (P) and the Ideal gas constant (R), and then multiplying by the Amount of gas (n), so we get: \(V = \frac{nRT}{P}\)

Invoke the Condition of Constant Pressure

As we are deriving Charles's gas law, which states that the volume of a fixed amount of gas is directly proportional to its temperature at constant pressure. Therefore, we assume that pressure and the amount of gas remains constant. Consequently, we can treat n, R, and P as constants in our equation: \(V = kT\) Where: \(k\) = Constant representing the product \(\frac{nR}{P}\)

State Charles's Gas Law

From Step 3, we derived the relationship between the volume of the gas and its temperature at constant pressure: \(V = kT\) This relationship is known as Charles's Gas Law, where the volume of the gas is directly proportional to its temperature at a constant pressure.

Key concepts

Charles's Law

Charles's Law is a fundamental concept in the study of gases. It describes how gases tend to expand when heated. In essence, it states that for a given amount of gas at constant pressure, the volume is directly proportional to its temperature in Kelvin. This means if the temperature of a gas is increased, its volume will increase likewise, assuming pressure remains unchanged. Mathematically, Charles's Law can be expressed as:

• \( V = kT \)

Here, \( V \) is the volume of the gas, \( T \) is the temperature, and \( k \) is a constant. This law visualizes the behavior of gases under thermal changes and is typically graphically represented as a straight line in a temperature (x-axis) versus volume (y-axis) plot. Understanding Charles's Law helps to predict how a gas will react to temperature changes, critical in fields like meteorology and engineering.

Gas Laws

Gas Laws are a set of rules that describe the behavior of gases and govern the relationships between pressure, volume, temperature, and the number of moles of a gas. These relationships are crucial for multiple practical applications, from predicting weather patterns to industrial chemical reactions. The Gas Laws include several important individual laws:

• Boyle's Law: Describes the inverse relationship between pressure and volume under constant temperature for a fixed amount of gas.
• Charles's Law: (As discussed above) Explains the direct relationship between volume and temperature at constant pressure.
• Avogadro's Law: States that equal volumes of gases at the same temperature and pressure contain equal numbers of molecules.
• Combined Gas Law: Integrates Boyle's, Charles's, and Avogadro's laws into a single equation useful for calculations.

These laws work together to form the basis for the Ideal Gas Law, which provides a comprehensive description of a gas's state under varying conditions.

Volume-Temperature Relationship

The volume-temperature relationship in gases is a key principle articulated in Charles's Law. It posits that the volume of a fixed mass of gas is directly proportional to its temperature, measured in Kelvin, when pressure remains constant. This principle can be explained as:

• As a gas is heated, its molecules gain kinetic energy and move more rapidly.
• This increased movement causes the molecules to collide with the walls of their container more frequently and with more force, thus expanding the container if it is flexible.
• Conversely, cooling a gas reduces the energy and movement of its molecules, leading to a decrease in volume.

This relationship is critical for understanding why materials expand with heat and contract with cold, which is an essential consideration in many practical fields, such as materials science and engineering, where temperature fluctuations are a concern.

Constant Pressure

The concept of constant pressure is an essential condition for applying Charles's Law. Constant pressure means that during the experiment or process, the pressure of the gas does not change, allowing us to observe the direct volume-temperature relationship. When pressure is held constant:

• Changes in temperature will affect the volume but not the pressure of the gas.
• This makes it easier to measure and demonstrate the variations in volume caused by thermal changes, as described by Charles's Law.
• In practice, maintaining constant pressure might involve systems that allow the gas to expand or contract freely, such as a gas-filled balloon or piston.

Understanding the idea of constant pressure helps students better grasp the fundamental principles of gas behavior and apply them correctly in both theoretical calculations and real-world scenarios.