Problem 40 - Chemistry for Allied Health

Question

A high altitude balloon is filled with \(1.41 \times 10^{4} \mathrm{~L}\) of hydrogen at a temperature of \(21^{\circ} \mathrm{C}\) and a pressure of 745 torr. What is the volume of the balloon at a height of \(20 \mathrm{~km}\), where the temperature is \(-48^{\circ} \mathrm{C}\) and the pressure is \(63.1\) torr?

Step-by-Step Solution

Answer

The volume of the balloon at 20 km height is approximately \(3.24 \times 10^{4} \text{ L}\).

Step 1: Convert Temperatures to Kelvin

We must convert the given temperatures from Celsius to Kelvin for use in the gas law equations.The initial temperature \(T_1 = 21^{\circ} \mathrm{C}\). To convert to Kelvin, add 273.15: \(T_1 = 21 + 273.15 = 294.15 \mathrm{~K}\).The final temperature \(T_2 = -48^{\circ} \mathrm{C}\). Similarly, \(T_2 = -48 + 273.15 = 225.15 \mathrm{~K}\).

Step 2: Understand the Problem

We're asked to find the final volume \(V_2\) of a gas using the Ideal Gas Law variables. We have initial and final conditions of pressure, volume, and temperature, so we'll use the combined gas law formula.

Step 3: Apply the Combined Gas Law Formula

The combined gas law formula is \(\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\). Solve for the unknown \(V_2\). Substituting the known values: \[\frac{745 \times 1.41 \times 10^{4}}{294.15} = \frac{63.1 \times V_2}{225.15}.\]

Step 4: Solve for Final Volume \(V_2\)

Rearrange the formula to solve for \(V_2\):\[ V_2 = \frac{(P_1 \cdot V_1 \cdot T_2)}{(T_1 \cdot P_2)}.\]Substitute the values: \[ V_2 = \frac{745 \times 1.41 \times 10^{4} \times 225.15}{294.15 \times 63.1} \approx 3.24 \times 10^{4} \text{ L}.\]

Step 5: Verify and Conclude

Double-check calculations and ensure all unit conversions were done correctly. The calculations confirm that the final volume \(V_2\) is approximately \(3.24 \times 10^{4} \text{ L}\).

Key Concepts

High Altitude Balloon

High altitude balloons are fascinating tools used for scientific research and meteorological studies. When filled with gases like hydrogen or helium, they can ascend to great heights in the atmosphere. Unlike a party balloon, a high altitude balloon can reach altitudes of up to 20 kilometers or more. The conditions at such altitudes are quite different from those near the Earth's surface, with much lower pressures and temperatures.
As a balloon rises higher, the surrounding pressure decreases, and it expands as a result. This expansion is directly influenced by the interactions between pressure, volume, and temperature, which is best explained through the lens of gas laws.

Temperature Conversion

Temperature conversion is a critical step when working with gas equations. The Kelvin scale is particularly important in chemistry because it starts at absolute zero, making it suitable for scientific calculations.

• To convert from Celsius to Kelvin, simply add 273.15 to the Celsius temperature.
• For example, converting from 21°C to Kelvin gives you 294.15 K.
• This conversion ensures consistency in calculations involving gas laws, as Kelvin is the standard unit of temperature used.

Neglecting to convert temperatures can result in incorrect outcomes when solving gas law problems.

Ideal Gas Law

The ideal gas law is a fundamental equation in understanding how gases behave under various conditions. It combines several gas laws, providing a comprehensive framework:

• The formula is typically expressed as: PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature.
• This law assumes ideal conditions, meaning the gas particles do not interact and move randomly.
• The combined gas law, which comes from rearranging the ideal gas law, is especially useful when dealing only with pressure, volume, and temperature changes.

By using the combined gas law \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \), we can calculate the new volume of a gas after a change in conditions.

Pressure-Volume Relationship

The pressure-volume relationship is a key concept in understanding gas behavior. It is governed by Boyle's Law, which is one of the components of the ideal gas law.

• Boyle's Law states that at constant temperature, the pressure and volume of a gas are inversely proportional. This means when pressure increases, volume decreases, and vice versa.
• In the combined gas law, these relationships allow us to predict how a gas will expand or compress when subjected to changes in environmental conditions.
• For a high altitude balloon, as it ascends and the external pressure drops, the volume must increase, highlighting the practical application of Boyle's Law.

Understanding these relationships is crucial for correctly predicting and analyzing real-world gas behavior changes, especially in scenarios like a rising balloon.