Problem 35 - Chemistry

Question

A gas-filled balloon having a volume of \(2.50 \mathrm{~L}\) at 1.2 atm and \(25^{\circ} \mathrm{C}\) is allowed to rise to the stratosphere (about \(30 \mathrm{~km}\) above the surface of Earth), where the temperature and pressure are \(-23^{\circ} \mathrm{C}\) and \(3.00 \times 10^{-3}\) atm, respectively. Calculate the final volume of the balloon.

Step-by-Step Solution

Answer

The final volume of the balloon is 3000 L.

Step 1: Exhibit Given Data

From the problem, initial pressure (P1) = 1.2 atm, initial volume (V1) = 2.50 L, initial temperature (T1) = 25 degree Celsius, final pressure (P2) = 3.00 x 10^-3 atm, final temperature (T2) = -23 degree Celsius. The final volume V2 is what we want to find.

Step 2: Convert Temperatures to Kelvin

Given temperatures have to be converted to absolute scale (Kelvin) before being included in the combined gas law equation. The formula to convert Celsius to Kelvin is K = C + 273.15. So, T1 = 25 + 273.15 = 298.15 K and T2 = -23 + 273.15 = 250.15 K.

Step 3: Apply Combined Gas Law

The combined gas law is given by P1 V1 / T1 = P2 V2 / T2. We'll insert our known values into the equation and solve for V2, the final volume. So, we have (1.2 atm * 2.50 L) / 298.15 K = (3.00 x 10^-3 atm * V2) / 250.15 K.

Step 4: Calculate Final Volume

Solving the equation for V2, we get V2 = (1.2 atm * 2.50 L * 250.15 K) / (3.00 x 10^-3 atm * 298.15 K) = 3000 L. Note, that atm units cancel out and K units cancel out. So, the final volume is in liters.

Key Concepts

Gas-Filled Balloon

A gas-filled balloon is a captivating example illustrating the behavior of gases under various environmental conditions. When a balloon is filled with gas, it contains a finite number of gas particles, enclosed within a flexible material that allows the volume of the balloon to expand or contract. The state of the gas inside the balloon can be described using variables such as volume, pressure, and temperature, which are all interconnected.
When a gas-filled balloon rises from the ground towards the stratosphere, it encounters changes in atmospheric pressure and temperature. These changes can cause the gas inside the balloon to behave differently by either expanding, contracting, or maintaining its state, depending on the external conditions. This is where understanding the combined gas law becomes essential, as it helps to predict how the volume of the gas will change in response to the varying pressure and temperature it experiences as it ascends.

Volume and Pressure Relationship

The volume and pressure relationship of a gas is a fundamental aspect of the combined gas law, which states that the volume of a gas is inversely proportional to its pressure when the temperature is constant. This means that as the pressure on a gas increases, its volume decreases, and vice versa.
For example, when a gas-filled balloon rises into the atmosphere, the pressure around it decreases. According to the combined gas law, if the temperature were constant (which it is not in this case), the balloon would expand to maintain the same number of gas particles within a larger space. However, in real situations, temperature also changes with altitude, adding a layer of complexity to the gas' behavior. The combined gas law seamlessly integrates this complexity by considering the variables of pressure, volume, and temperature simultaneously, providing a more comprehensive understanding of how gases respond to different conditions.

Temperature Conversion Kelvin

Temperature plays a crucial role in understanding gas behavior and is a key component of the combined gas law equation. However, it's important to note that all gas law calculations require temperature to be expressed in Kelvin, which is the absolute temperature scale. The reason for using Kelvin is that it begins at absolute zero, the theoretical point where particles have minimal vibrational motion.
To convert Celsius to Kelvin, we add 273.15 to the Celsius temperature. This adjustment ensures that the temperature values used in the gas law equations are positive and proportional to the absolute number of particles in motion. In our balloon example, the temperature conversion from Celsius to Kelvin for the initial and final temperatures is critical to finding the final volume of the balloon. Without this step, the combined gas law calculation would be incorrect as the equation depends on an absolute temperature scale, making the Kelvin conversion an indispensable aspect of understanding and applying the combined gas law.