Question

A weather balloon is inflated to a volume of 28.5 \(\mathrm{L}\) at a pressure of 748 \(\mathrm{mm} \mathrm{Hg}\) and a temperature of \(28.0^{\circ} \mathrm{C}\) .The balloon rises in the atmosphere to an altitude of approximately \(25,000\) feet, where the pressure is 385 \(\mathrm{mm} \mathrm{Hg}\) and the temperature is \(-15.0^{\circ} \mathrm{C}\) . Assuming the balloon can freely expand, calculate the volume of the balloon at this altitude.

Short answer

The volume of the balloon at an altitude of 25,000 feet is approximately 47.9 L.

Step-by-step solution

Convert temperatures to Kelvin

Since the gas law requires temperature to be in Kelvin, convert both temperatures from Celsius to Kelvin using the formula: Kelvin = Celsius + 273.15. For the initial temperature, T1 = 28.0°C + 273.15 = 301.15 K. For the final temperature, T2 = -15.0°C + 273.15 = 258.15 K.

Use the Combined Gas Law

Apply the Combined Gas Law, which is expressed as (P1 * V1) / T1 = (P2 * V2) / T2. Here, P1 and V1 are the initial pressure and volume, and P2 and V2 are the final pressure and volume. Solve the equation for V2 to find the final volume of the balloon.

Solve for the final volume V2

Rearrange the Combined Gas Law to solve for V2 gives V2 = V1 * (P1/P2) * (T2/T1). Substituting the known values, V2 = 28.5 L * (748 mmHg / 385 mmHg) * (258.15 K / 301.15 K).

Calculate the final volume

Perform the calculation from the previous step to get V2 = 28.5 L * (748/385) * (258.15/301.15) = 28.5 L * 1.942857 * 0.85737 = 47.876 L (rounded to three significant digits).

Key concepts

Gas Law Calculations

Gas law calculations are essential in understanding the relationship between pressure, volume, and temperature of a gas sample. Consider the weather balloon problem where we seek to calculate the change in volume as it ascends in the atmosphere. To tackle such problems, we make use of the Combined Gas Law, which unifies Boyle's, Charles', and Gay-Lussac's laws.

This law is expressed mathematically as \( (P1 \cdot V1) / T1 = (P2 \cdot V2) / T2 \), where P stands for pressure, V for volume, and T for temperature. The subscripts 1 and 2 represent the initial and final states of the gas, respectively. It's crucial to realize that these calculations assume the amount of gas and the gas itself remain constant.

Importance of Gas Constants

When performing the calculations, remember that pressure and temperature must be in consistent units - pressure in atmospheres, Pascals, or mmHg, and temperature in Kelvin. It's these unwavering regulations of standard units that allow for accurate and reliable results in gas law calculations.

Converting Celsius to Kelvin

In gas law problems such as the weather balloon scenario, temperature must always be converted to the Kelvin scale. This is because Kelvin is the absolute temperature scale and the gas laws are directly derived from the kinetic theory which uses this scale.

To convert Celsius to Kelvin, we use the simple formula: \( \text{K} = \text{C} + 273.15 \). For instance, the initial temperature of the balloon, at 28.0°C, when converted to Kelvin becomes \( 28.0^\circ \text{C} + 273.15 = 301.15 \text{ K} \).

Understanding Kelvin

The Kelvin scale starts from absolute zero, the coldest possible temperature where particles theoretically stop moving. This is why Kelvin allows for a proportionate mathematical relationship in gas law equations, which is not possible with Celsius or Fahrenheit due to their arbitrary zero points.

Behavior of Gases under Different Temperatures and Pressures

The behavior of gases under various temperatures and pressures is key to predicting how they will act in different situations - like our weather balloon's ascent. Gases expand when heated and contract when cooled, also responding predictably to changes in pressure.

As seen in the weather balloon example, as it rises and encounters lower pressure and temperatures, the volume adjusts accordingly. This is due to gas particles moving faster at higher temperatures and exerting more pressure on their container, which leads to expansion, and the reverse when they cool down.

Gas Laws in Action

In our case, the balloon's initial conditions are a higher pressure and temperature; therefore, it has a smaller volume. When it reaches higher altitude with lower pressure and temperature, the volume increases. This principle is essential not just for this problem but also in various industrial and natural processes involving gases.