Problem 44 - Chemistry A Molecular Approach

Question

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

Step-by-Step Solution

Answer

The volume of the balloon at the altitude of 25,000 ft would be \(49.9 \mathrm{L}\).

Step 1: Convert Temperatures to Kelvin

Convert the given temperatures from Celsius to Kelvin using the formula: Kelvin = Celsius + 273.15. For the initial temperature, convert 28.0°C to K. Similarly, for the final temperature, convert -15.0°C to K.

Step 2: Apply Ideal Gas Law

Use the Ideal Gas Law in the form of the combined gas law to express the initial state and final state of the gas: \(P_1V_1/T_1 = P_2V_2/T_2\), where \(P\) is pressure, \(V\) is volume, and \(T\) is temperature.

Step 3: Rearrange the Equation to Solve for Final Volume

Rearrange the combined gas law to solve for \(V_2\): \(V_2 = (P_1V_1/T_1) \cdot (T_2/P_2)\). Plug in the initial conditions and solve for the new volume.

Step 4: Calculate New Volume

Insert all the known values into the rearranged formula and calculate the volume of the balloon at the higher altitude.

Key Concepts

Combined Gas Law

The combined gas law is a fundamental concept in chemistry and physics, linking pressure, volume, and temperature of a gas in a single, versatile equation. It combines Charles's Law, Boyle's Law, and Gay-Lussac's Law, and is expressed as \( \frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2} \), where \(P_1\) and \(P_2\) are the pressures at states 1 and 2, \(V_1\) and \(V_2\) are the volumes at states 1 and 2, and \(T_1\) and \(T_2\) are the temperatures at states 1 and 2, respectively.

For instance, when a weather balloon ascends, the combined gas law can be used to predict how its volume will change in response to the corresponding changes in atmospheric pressure and temperature. Remember, temperature must always be in Kelvin for these calculations to reflect the correct physical behavior of gases.

Converting Celsius to Kelvin

Temperature conversions are essential in gas law calculations since all gas laws require the absolute temperature in Kelvin (K). The Kelvin scale is the SI unit for temperature, and it starts at absolute zero, unlike the Celsius scale, where 0 degrees is set at the freezing point of water. To convert Celsius to Kelvin, you simply add 273.15 to the Celsius temperature using the formula: \(\text{Kelvin} = \text{Celsius} + 273.15\).

For our weather balloon example, converting the initial temperature \(\text{28.0}^\circ \text{C}\) and the final temperature \(\text{-15.0}^\circ \text{C}\) to Kelvin ensures that the subsequent calculations quantitatively describe the behavior of the gas inside the balloon with accuracy.

Gas Law Calculations

Performing gas law calculations allows for the determination of an unknown quantity (pressure, volume, or temperature) of a gas when the other quantities are known or can be measured. In the context of solving for the expanded volume of a weather balloon at high altitudes, after converting temperatures to Kelvin, we use the rearranged combined gas law to solve for \(V_2\) as follows: \(V_2 = \frac{P_1V_1T_2}{T_1P_2}\).

With all the values plugged in (and ensuring units for pressure and volume remain consistent), students can calculate the new volume, effectively applying the combined gas law to a real-world scenario.

Atmospheric Pressure Changes

Atmospheric pressure decreases with altitude because the atmosphere is less dense the higher you go. This is due to the gravitational force being stronger closer to the surface of the Earth, pulling more air molecules toward it and creating more dense layers of atmosphere at lower altitudes.

In the case of the weather balloon, as it rises, it encounters lower atmospheric pressure, which allows the balloon to expand. Understanding how pressure changes with altitude is crucial when applying gas laws, as it is often a variable that must be taken into account, as seen with the balloon's volume adjustment at 25,000 feet compared to ground level.