Problem 56 - Chemistry Matter and Change

Question

Weather Balloons A weather balloon is filled with helium that occupies a volume of \(5.00 \times 10^{4} \mathrm{L}\) at 0.995 atm and \(32.0^{\circ} \mathrm{C}\) . After it is released, it rises to a location where the pressure is 0.720 atm and the temperature is \(-12.0^{\circ} \mathrm{C}\) . What is the volume of the balloon at the new location?

Step-by-Step Solution

Answer

The volume of the balloon at the new location is approximately 59273.53 liters.

Step 1: Convert Temperatures to Kelvin

First, convert the given temperatures from degrees Celsius to Kelvin because gas laws require temperatures in Kelvin. The formula to convert Celsius to Kelvin is:\[ K = ^{\circ}C + 273.15 \]For the initial temperature and the final temperature:Initial Temperature: \(32.0^{\circ}C = 32.0 + 273.15 = 305.15 K\)Final Temperature: \(-12.0^{\circ}C = -12.0 + 273.15 = 261.15 K\)

Step 2: Use the Combined Gas Law

The combined gas law relates pressure, volume, and temperature between two states of a gas. It is given by the formula:\[ \frac{P_1 \cdot V_1}{T_1} = \frac{P_2 \cdot V_2}{T_2} \]Where - \(P_1\), \(V_1\), and \(T_1\) are the initial pressure, volume, and temperature, - \(P_2\), \(V_2\), and \(T_2\) are the pressure, volume, and temperature at the new location.

Step 3: Plug in Known Values

Substitute the known values into the combined gas law equation:\[\frac{0.995 \cdot 5.00 \times 10^4}{305.15} = \frac{0.720 \cdot V_2}{261.15} \]

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

To find \(V_2\), rearrange the equation:\[ V_2 = \frac{0.995 \cdot 5.00 \times 10^4 \cdot 261.15}{305.15 \cdot 0.720} \]Now, calculate \(V_2\):\[V_2 = \frac{13023750}{219.708}\approx 59273.53 \text{ L}\]

Step 5: Conclusion

The volume of the weather balloon at the new location is approximately 59273.53 liters.

Key Concepts

Temperature Conversion

Understanding temperature conversion is crucial when dealing with gas laws because most equations require temperatures to be in Kelvin rather than Celsius. The Kelvin scale is the SI unit for thermodynamic temperature measurement, eliminating negative values, which are not practical in many physical calculations.

To convert from Celsius to Kelvin, simply add 273.15 to the Celsius temperature. For example:

• The initial temperature in the exercise is 32.0 °C. By adding 273.15, we convert it to 305.15 K.
• Similarly, the final temperature is -12.0 °C, which becomes 261.15 K after conversion.

Remember, using Kelvin ensures that all calculations in gas law equations are straightforward and logically consistent.

Volume Calculation

Volume calculation is a central aspect of applying the gas laws because it often changes based on conditions such as pressure and temperature. In the exercise, the volume of the balloon at different altitudes is calculated to understand how it expands as it rises into the atmosphere.

Here, we determine the final volume of the weather balloon using the combined gas law.

• The initial volume is given as 5.00 × 10⁴ L.
• Using the combined gas law formula and rearranging for the unknown variable helps us determine the new volume.

The process involves substituting known variables into the formula and solving for the unknown volume, which illustrates how gases behave under different atmospheric conditions.

Gas Laws

Gas laws describe the behavior of gases in relation to pressure, temperature, and volume. The Combined Gas Law is especially pertinent as it combines three fundamental gas laws: Boyle's Law, Charles's Law, and Gay-Lussac's Law.

This law is represented as \[\frac{P_1 \cdot V_1}{T_1} = \frac{P_2 \cdot V_2}{T_2}\]where:

• \(P_1\) and \(P_2\) are the initial and final pressures,
• \(V_1\) and \(V_2\) are the initial and final volumes,
• \(T_1\) and \(T_2\) are the initial and final temperatures in Kelvin.

By plugging the known values into this equation, you can compute the unknown variables, demonstrating the predictable ways gases can expand or contract depending on external conditions.