Problem 14 - Chemistry and Chemical Reactivity

Question

A helium-filled balloon of the type used in longdistance flying contains \(420,000 \mathrm{ft}^{3}\left(1.2 \times 10^{7} \mathrm{L}\right)\) of helium. Suppose you fill the balloon with helium on the ground, where the pressure is \(737 \mathrm{mm} \mathrm{Hg}\) and the temperature is \(16.0^{\circ} \mathrm{C}\) When the balloon ascends to a height of 2 miles, where the pressure is only \(600 . \mathrm{mm}\) Hg and the temperature is \(-33^{\circ} \mathrm{C},\) what volume is occupied by the helium gas? Assume the pressure inside the balloon matches the external pressure.

Step-by-Step Solution

Answer

The helium balloon's volume at 2 miles altitude is approximately 498,006 cubic feet.

Step 1: Identify Initial Conditions

First, note the initial conditions at the ground level: the initial volume of helium is \( V_1 = 420,000 \text{ ft}^3 \), the initial pressure is \( P_1 = 737 \text{ mm Hg} \), and the initial temperature is \( T_1 = 16.0^{\circ} C = 289.15 \text{ K} \).

Step 2: Identify Final Conditions

Next, consider the final conditions at a height of 2 miles: the final pressure is \( P_2 = 600 \text{ mm Hg} \), and the final temperature is \( T_2 = -33^{\circ} C = 240.15 \text{ K} \).

Step 3: Apply the Combined Gas Law

The combined gas law \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \) relates pressure, volume, and temperature. Substitute the known values into this formula to solve for the final volume \( V_2 \).

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

Rearrange the combined gas law to solve for \( V_2 \): \[ V_2 = \frac{P_1 V_1 T_2}{P_2 T_1}. \] Substitute the values: \[ V_2 = \frac{737 \text{ mm Hg} \times 420,000 \text{ ft}^3 \times 240.15 \text{ K}}{600 \text{ mm Hg} \times 289.15 \text{ K}}. \] Calculate \( V_2 \).

Step 5: Calculate and Interpret the Final Volume

Now, perform the calculation: \[ V_2 = \frac{737 \times 420,000 \times 240.15}{600 \times 289.15} \approx 498,006 \text{ ft}^3. \]Thus, at the higher altitude, the balloon's volume of helium increases to approximately 498,006 cubic feet.

Key Concepts

Gas Laws

The gas laws are fundamental principles that describe the behavior of gases in relation to pressure, volume, and temperature. These laws were formulated over centuries starting with Boyle's and Charles' laws, leading to the development of the Combined Gas Law.

• Boyle's Law: It states that for a given mass of gas at constant temperature, the volume of the gas varies inversely with its pressure.
• Charles' Law: This law shows that the volume of gas is directly proportional to its temperature at constant pressure.
• Gay-Lussac's Law: It indicates the direct relationship between temperature and pressure at a constant volume.

Understanding these individual laws helps in comprehending the Combined Gas Law. Combined Gas Law integrates them into a single equation: \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \). It demonstrates how the pressure, volume, and temperature of a gas are interdependent. This equation is a valuable tool in solving problems where any two of the gas properties change, such as in our helium balloon scenario.

Helium Balloon

A helium balloon is a perfect example to illustrate gas laws in a real-world application. Helium, a non-reactive and lighter-than-air gas, is used because it allows balloons to float. In the context of gas laws, helium balloons demonstrate how changes in external conditions alter the pressure, volume, and temperature of the gas inside.
When you fill a balloon with helium, the gas particles inside push against the balloon’s walls, creating pressure. As a balloon rises, differences in external air pressure and temperature will affect the behavior of the gas inside. Helium balloons thus become a great educational tool to visualize the interactions dictated by the Combined Gas Law.
In our exercise, the balloon starts at ground level and ascends to higher altitudes, showcasing how changes in pressure and temperature affect the gas volume.

Pressure-Volume Relationship

The pressure-volume relationship is a core concept in understanding gas behavior, essentially communicated by Boyle's Law. It states that for a given mass and constant temperature, the volume of a gas is inversely proportional to its pressure.
In the exercise, the balloon's helium gas volume increases as it rises and the external pressure decreases. This behavior is intuitive:

• If the pressure on a gas decreases, the gas expands, enlarging its volume.
• Conversely, an increase in pressure would compress the gas, thus lowering the volume.

The exercise illustrates this phenomenon as the balloon ascends to a lower-pressure environment, resulting in an increase in the helium volume from 420,000 cubic feet to approximately 498,006 cubic feet. This clearly demonstrates the inverse pressure-volume relationship.

Temperature Effects on Gases

Temperature changes have significant effects on gas behavior. This is explained by Charles' Law, which shows that gas volume is directly proportional to its temperature when pressure is constant.
In our balloon exercise, as the temperature decreases when the balloon rises, we see this effect in action:

• At the ground level, the temperature is 16°C (or 289.15 K), and as the balloon rises to an altitude of two miles, the temperature drops to -33°C (or 240.15 K).
• The decrease in temperature means that at lower pressures, gas molecules have less kinetic energy, which could cause a reduction in volume if isolated. However, the reduced pressure at altitude allows expansion despite the colder temperature.

Understanding how temperature affects gas expansion and contraction is crucial, especially in practical scenarios like ballooning or any application involving varying environmental conditions.