Question

Starting from the ideal gas law, prove that the volume of a mole of gas is directly proportional to the absolute temperature at constant pressure (Charles's law).

Short answer

The volume of a gas at constant pressure is directly proportional to its absolute temperature.

Step-by-step solution

- Understand the Ideal Gas Law

The Ideal Gas Law is represented by the formula \( PV = nRT \), where: \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the absolute temperature. We need to show how volume is related to temperature at constant pressure for one mole of gas.

- Set Conditions for Charles's Law

According to Charles's Law, the volume of a gas is proportional to its temperature when pressure is constant. We assume \( n = 1 \) mole, as we are considering the volume of one mole of gas and pressure \( P \) remains constant.

- Simplify the Ideal Gas Law

With the conditions set in Step 2, substitute \( n = 1 \) into the Ideal Gas Law equation: \( PV = RT \). This equation can be rewritten as \( V = \frac{RT}{P} \), showing that \( V \) is directly proportional to \( T \), as \( R \) and \( P \) are constants.

- Conclude with Charles's Law

From \( V = \frac{RT}{P} \), we observe that when \( P \) is constant, changes in \( T \) will result in proportional changes in \( V \). This proves that at constant pressure, the volume of a mole of gas is directly proportional to its absolute temperature, which is the essence of Charles's Law.

Key concepts

Ideal Gas Law

The Ideal Gas Law is an essential concept in chemistry and physics that describes how gases behave under various conditions. It is represented by the formula \( PV = nRT \), which combines several simpler gas laws into one comprehensive equation. This formula connects four important properties of gases:

• Pressure (\( P \)): The force exerted by gas particles colliding with the walls of their container.
• Volume (\( V \)): The space that the gas occupies.
• Moles (\( n \)): The amount of substance, typically measured in moles.
• Temperature (\( T \)): The measure of kinetic energy of particles, always in Kelvin for this equation.
• Ideal Gas Constant (\( R \)): A constant which is typically approximately 8.314 J/(mol·K).

Together, these variables describe the state of a sample of gas. By holding certain variables constant, you can deduce how changing one variable affects the others. This is particularly useful for deriving specific gas laws, such as Charles's Law, by starting from this foundational equation.

volume and temperature relationship

The volume and temperature relationship is a key aspect of understanding how gases behave. According to Charles's Law, when the pressure is kept constant, the volume of a gas is directly proportional to its temperature in Kelvin. This means that if you increase the temperature of the gas, its volume will increase as well, and vice versa.

• Direct Proportionality: Given by the equation \( V = kT \), where \( k \) is a constant for a given amount of gas at constant pressure.
• Absolute Temperature: It is crucial to use Kelvin, not Celsius, since the law relies on absolute temperatures to ensure the ratio remains constant.

In the context of the Ideal Gas Law, if the number of moles and pressure remain constant, we can rearrange the equation \( PV = nRT \) as \( V = \frac{RT}{P} \). This form clearly shows how volume changes proportionately with temperature, illustrating Charles's Law in action.

constant pressure

Constant pressure is a fundamental condition for applying Charles's Law to relate volume and temperature. In many experiments and real-world applications, pressure is kept steady to study the specific effects of temperature on a gas's volume without introducing additional variables.

• Pressure as a Control Parameter: By fixing the pressure, scientists can isolate the two variables of interest (volume and temperature) to better understand their relationship.
• Implication in Experiments: When pressure is constant, changes in temperature directly lead to proportional changes in volume according to Charles's Law.

Maintaining constant pressure allows the rearrangement of the Ideal Gas Law to clearly observe the linear relationship between temperature and volume. Understanding these controlled environments is crucial for accurately predicting how a gas will respond to temperature changes. This makes constant pressure experiments a cornerstone in gas law studies and applications.