Question

A child receives a balloon filled with 2.30 L of helium from a vendor at an amusement park. The temperature outside is 311 K. What will the volume of the balloon be when the child brings it home to an air-conditioned house at 295 K? Assume that the pressure stays the same.

Short answer

The volume of the balloon when brought home will be approximately 2.18 L.

Step-by-step solution

Identify the Known Variables

The initial volume of the balloon is 2.30 L. The initial temperature (outside) is 311 K, and the final temperature (inside the house) is 295 K. The pressure is assumed to remain constant.

Apply Charles's Law

Charles's Law states that, for a given amount of gas at constant pressure, the volume is directly proportional to its temperature in kelvin. Mathematically, it is expressed as: \ \ \ V_1 / T_1 = V_2 / T_2 \ \ Here, \(V_1 = 2.30 \ L, T_1 = 311 \ K, T_2 = 295 \ K\)

Solve for the Final Volume

Rearrange Charles's Law equation to solve for the final volume \(V_2\): \ \ V_2 = V_1 \cdot \frac{T_2}{T_1} \ \ Substitute the values: \ \ V_2 = 2.30 \ \frac{295}{311} \ \ Calculate the final volume: \ \ V_2 \approx 2.18 L

Key concepts

Charles's Law

Charles's Law is one of the fundamental principles in chemistry related to gas behavior. It states that the volume of a given mass of gas is directly proportional to its absolute temperature, as long as the pressure remains constant. This means that if you increase the temperature of the gas, its volume will also increase and vice versa.

The relationship is mathematically expressed as: \[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \] Here, \( V_1 \) and \( T_1 \) are the initial volume and temperature, and \( V_2 \) and \( T_2 \) are the final volume and temperature, respectively. Absolute temperature is measured in Kelvin, and this law only holds true when the pressure does not change.

Understanding Charles's Law is vital because it allows us to predict how the volume of a gas will react to changes in temperature, which has practical applications in everyday situations and scientific experiments.

Gas Laws

Gas laws are a set of laws that describe how gases behave with respect to pressure, volume, and temperature. These laws help scientists and engineers to understand and predict the behavior of gases in various conditions. The most well-known gas laws include:

* Boyle's Law: States that the volume of a gas is inversely proportional to its pressure, at constant temperature.
* Charles's Law: As explained earlier, it states that the volume of a gas is directly proportional to its temperature, at constant pressure.
* Avogadro's Law: States that the volume of a gas is directly proportional to the number of moles of gas, at constant temperature and pressure.
* Gay-Lussac's Law: States that the pressure of a gas is directly proportional to its temperature, at constant volume.

These laws form the basis for understanding complex gas behaviors and are essential for various applications in science, industry, and everyday life.

Temperature and Volume Relationship

The relationship between temperature and volume, as described by Charles's Law, is a crucial concept in understanding gas behavior. When the temperature of a gas increases, the kinetic energy of the gas molecules increases as well. This leads to more frequent collisions with the walls of their container, causing the gas to expand and occupy a larger volume.

Conversely, when the temperature decreases, the kinetic energy of the gas molecules decreases, resulting in fewer collisions with the container’s walls. This causes the gas to contract and occupy a smaller volume.

For example, in the provided exercise, the initial conditions are a volume of 2.30 L at an outside temperature of 311 K. When the balloon is brought inside a house where the temperature is 295 K, Charles's Law predicts the new volume of the balloon. Using the equation \[ V_2 = V_1 \frac{T_2}{T_1} \] and substituting the values: \[ V_2 = 2.30 \frac{295}{311} \] gives us an approximate final volume of 2.18 L.

This demonstrates how a decrease in temperature results in a decrease in the volume of gas, maintaining the pressure constant.