Question

A \(36.4-\mathrm{L}\) volume of methane gas is heated from \(25^{\circ} \mathrm{C}\) to \(88^{\circ} \mathrm{C}\) at constant pressure. What is the final volume of the gas?

Short answer

The final volume of the gas is approximately 44.06 L.

Step-by-step solution

Convert Celsius to Kelvin

In order to apply Charles's Law, the temperatures must be in Kelvin. The formula to convert Celsius to Kelvin is \(K = C + 273.15\). Thus, the initial and final temperatures in Kelvin are :\(T_{1} = 25^{\circ}C + 273.15 = 298.15 K\)\(T_{2} = 88^{\circ}C + 273.15 =361.15 K\)

Apply Charles's Law

Charles's Law states that the volume of a gas is directly proportional to its absolute temperature when pressure is kept constant. In mathematical terms, this is expressed as \(V_{1}/T_{1} = V_{2}/T_{2}\) where \(V_{1}\) and \(T_{1}\) are the initial volume and temperature, and \(V_{2}\) and \(T_{2}\) are the final volume and temperature. Substituting in the known values, we get: \(36.4 L / 298.15 K = V_{2} / 361.15 K\).

Solve for Final Volume

Cross multiply and solve for \(V_{2}\).When you cross multiply, you get: \(36.4 L * 361.15 K = V_{2} * 298.15 K\). Solving for \(V_{2}\) gives: \(V_{2} = (36.4 L * 361.15 K)/298.15 K\).

Key concepts

Understanding Gas Laws

Gas Laws are fundamental principles in chemistry and physics that describe the behavior of gases under various conditions of temperature, volume, and pressure. These laws are crucial in predicting how gases will react when subjected to changes in environmental conditions.

One of the most important gas laws is Charles's Law, which specifically focuses on how the volume of a gas changes with temperature, provided the pressure remains constant. The importance of Charles's Law cannot be overstated as it has practical applications in fields like meteorology, where it helps in understanding the behavior of atmospheric gases, and in automotive engineering, where it aids in tire pressure management.

Exploring the Volume-Temperature Relationship

Charles's Law illustrates the inherent volume-temperature relationship of gases. The volume of a gas is directly proportional to its absolute temperature, meaning if the temperature of a gas increases, so does its volume, and vice versa.

This relationship allows scientists and engineers to predict how a gas will expand or contract with temperature changes. In the classroom, understanding this principle helps students solve problems involving gas volumes and temperatures, like the example problem where a volume of methane gas expands as it is heated.

The Kelvin Temperature Scale

The Kelvin temperature scale is essential when working with gas laws because it measures the absolute temperature, which is necessary for Charles's Law calculations. Unlike Celsius or Fahrenheit, Kelvin does not have negative values, which aligns well with the concept of absolute temperature, where 0 K, or absolute zero, is theoretically the lowest temperature possible.

Using Kelvin removes the complexity that can arise from negative temperature values when calculating changes in gas properties and ensures accuracy when dealing with proportional relationships between volume and temperature in gases.