Problem 16 - Chemistry Matter and Change

Question

Analyze A weather balloon is released into the atmosphere. You know the initial volume, temperature, and air pressure. What information will you need to predict its volume when it reaches its final altitude? Which law would you use to calculate this volume?

Step-by-Step Solution

Answer

To predict the final volume of a weather balloon when it reaches its final altitude, you will need the final temperature (T₂) and final air pressure (P₂). You can use the Combined Gas Law, which is mathematically expressed as \(P₁V₁/T₁ = P₂V₂/T₂\). Rearrange this equation to solve for the final volume (V₂) as follows: \(V₂ = \frac{P₁V₁T₂}{T₁P₂}\). By substituting the known values, you can calculate the final volume of the weather balloon.

Step 1: Identify the given variables

The given variables are: 1. Initial volume (V₁) 2. Initial temperature (T₁) 3. Initial air pressure (P₁)

Step 2: Determine the needed variables

To predict the final volume of the weather balloon, we need the following variables: 1. Final temperature (T₂) 2. Final air pressure (P₂)

Step 3: Select the appropriate gas law

To determine the final volume of the weather balloon, we can use the Combined Gas Law, which combines Boyle's Law, Charles's Law, and Gay-Lussac's Law. This law states that the product of the initial pressure, volume, and temperature is directly proportional to the product of the final pressure, volume, and temperature when the amount of gas is held constant. Mathematically, it can be expressed as follows: \(P₁V₁/T₁ = P₂V₂/T₂\)

Step 4: Solve for the final volume (V₂)

Rearrange the Combined Gas Law equation to solve for the final volume (V₂): \(V₂ = \frac{P₁V₁T₂}{T₁P₂}\) Now, substitute the known values (initial volume, initial temperature, initial pressure, final temperature, and final pressure) into the equation and solve for V₂: \(V₂ = \frac{P₁V₁T₂}{T₁P₂}\) This will give us the final volume of the weather balloon when it reaches its final altitude.

Key Concepts

Weather Balloon Volume Prediction

Understanding how to predict the volume of a weather balloon as it ascends is a practical application of the gas laws. Weather balloons expand as they rise due to decreasing atmospheric pressure and potential changes in temperature. By knowing the initial volume, temperature, and pressure of the balloon, and predicting its final temperature and pressure at the targeted altitude, scientists can estimate how much the balloon will expand. To make this estimation, they employ the Combined Gas Law, which allows for the calculation of the unknown final volume (V₂).

Therefore, in addition to the initial conditions, you'll need to know or predict the final temperature (T₂) and the final pressure (P₂) that the balloon will face at its final altitude. These insights are crucial for high-altitude meteorological measurements and ensuring the weather balloon does not burst from overexpansion.

Gas Laws

The behavior of gases under various conditions of pressure, volume, and temperature is described by the gas laws. These laws help us to understand and predict how a gas will behave when one or more of its state variables are altered. For practical applications such as predicting the volume of a weather balloon at a certain altitude, these laws are crucial. The Combined Gas Law, which we are focusing on here, considers the simultaneous changes of pressure, volume, and temperature of a fixed amount of gas.

Boyle's Law

Understanding Pressure-Volume Relationship

Boyle's Law is one of the fundamental gas laws, stating that the pressure of a given mass of an ideal gas is inversely proportional to its volume when the temperature is held constant. Mathematically, Boyle's Law can be written as \( P_1V_1 = P_2V_2 \) where \( P_1 \) and \( P_2 \) are the initial and final pressures, and \( V_1 \) and \( V_2 \) are the initial and final volumes, respectively. This law is particularly useful when studying the behavior of gases being compressed or expanding at constant temperature.

Charles's Law

Temperature-Volume Relationship

Charles's Law explains how gases tend to expand when heated. The law states that the volume of a given amount of gas is directly proportional to its absolute temperature (in kelvins) if the pressure remains constant. The equation \( V_1/T_1 = V_2/T_2 \) encapsulates this relationship, with \( T_1 \) and \( T_2 \) representing the initial and final temperatures, and \( V_1 \) and \( V_2 \) representing the initial and final volumes. Charles's Law is vital for understanding the behaviors of a gas like the helium or hydrogen in a weather balloon as it ascends and the temperature changes.

Gay-Lussac's Law

Pressure-Temperature Relationship

Gay-Lussac's Law describes how pressure and temperature are related for a given mass of gas at constant volume. It states that the pressure exerted by a gas is directly proportional to its absolute temperature. The formula for this law is \( P_1/T_1 = P_2/T_2 \), where \( P_1 \) and \( P_2 \) are the initial and final pressures, and \( T_1 \) and \( T_2 \) are the initial and final temperatures, respectively. This law helps predict the changes in pressure of the trapped gas in a weather balloon due to temperature changes as it climbs to different altitudes.