Question

A gas-filled balloon having a volume of \(2.50 \mathrm{~L}\) at 1.2 atm and \(20^{\circ} \mathrm{C}\) is allowed to rise to the stratosphere (about \(30 \mathrm{~km}\) above the surface of Earth), where the temperature and pressure are \(-23^{\circ} \mathrm{C}\) and \(3.00 \times 10^{-3}\) atm, respectively. Calculate the final volume of the balloon.

Short answer

The final volume of the balloon is approximately 853.25 L.

Step-by-step solution

Understand the problem

We need to calculate the final volume of a gas-filled balloon as it rises to a different altitude with a change in temperature and pressure.

Identify Known Variables

Initial Volume, \(V_1 = 2.50 \mathrm{~L}\);Initial Pressure, \(P_1 = 1.2 \mathrm{~atm}\);Initial Temperature, \(T_1 = 20^{\circ} \mathrm{C} = 293.15 \mathrm{~K}\);Final Pressure, \(P_2 = 3.00 \times 10^{-3} \mathrm{~atm}\);Final Temperature, \(T_2 = -23^{\circ} \mathrm{C} = 250.15 \mathrm{~K}\).

Use the Ideal Gas Law Relation

We can relate the initial and final states of the gas using the combined gas law: \[\frac{P_1 \times V_1}{T_1} = \frac{P_2 \times V_2}{T_2}\]where \(V_2\) is the final volume we need to find.

Rearrange to Find Final Volume

Rearrange the formula to solve for \(V_2\): \[V_2 = V_1 \times \frac{P_1}{P_2} \times \frac{T_2}{T_1}\]

Substitute Known Values

Substitute the known values into the equation:\[V_2 = 2.50 \times \frac{1.2}{3.00 \times 10^{-3}} \times \frac{250.15}{293.15}\]

Calculate

First calculate the pressure ratio: \[\frac{1.2}{3.00 \times 10^{-3}} = 400\]Then calculate the temperature ratio: \[\frac{250.15}{293.15} \approx 0.853\]Now, multiply them together with the initial volume:\[V_2 = 2.50 \times 400 \times 0.853 \approx 853.25 \mathrm{~L}\]

Conclusion

The final volume of the balloon, when it reaches the stratosphere, is approximately 853.25 L.

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental principle in chemistry and physics. It describes the relationship between the pressure, volume, and temperature of an ideal gas. The equation is written as \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles of gas, \( R \) is the ideal gas constant, and \( T \) is temperature (in Kelvin).
For complex scenarios, such as changes happening simultaneously in pressure, volume, and temperature, we often use a derived equation called the combined gas law: \( \frac{P_1 \times V_1}{T_1} = \frac{P_2 \times V_2}{T_2} \). This simplifies calculations when comparing the initial and final states of a gas as it undergoes transformations.
While the ideal gas law offers a great approximation, remember it assumes no interactions between gas molecules and that the gas occupies no volume itself. Though real gases can deviate from this behavior under certain conditions, it generally provides a solid foundation for understanding gas behaviors.

Volume Calculation

Volume calculation in gases is all about understanding how volume changes with pressure and temperature. When we need to find the final volume of a gas under changing conditions, the combined gas law formula is invaluable: \( V_2 = V_1 \times \frac{P_1}{P_2} \times \frac{T_2}{T_1} \).
This equation represents a mathematical way to "track" how volume must adjust as pressure or temperature change. If pressure decreases and temperature increases, generally, the gas volume expands as molecules spread out more.
The solution showed us dividing the initial conditions by the final conditions. This process, called normalization, simplifies the variables, ensuring calculations align with proportionality traced from the physical gas laws.
Keep keen about unit consistency—temperatures must be in Kelvin, and pressure in matched units, for correct results with precision.

Altitude Effects on Gases

The altitude has a fascinating effect on gases due to changes in atmospheric pressure and temperature. As we climb higher, say into the stratosphere, atmospheric pressure drops significantly. This is because there are fewer air molecules pushing down from above.
In the context of our balloon, the volume expanded massively due to the decrease in ambient pressure. With less external pressure compressing the balloon, it inflates as the pressure inside tries to equilibrate with the exterior.
Similarly, temperature often drops as we ascend—though at stratospheric levels, other variables come into play, leading sometimes to warmer layers. Nonetheless, in this example, the colder temperatures slightly oppose the effect of decreased pressure, though not enough to negate the dominant expansion effect.

• Lower pressure higher altitudes = gas expansion
• Temperature decreases can slightly contract gases

Understanding these attributes explains a lot about weather balloons and high-altitude aircraft design.

Pressure and Temperature Changes

Pressure and temperature changes are crucial variables influencing gas behavior. When either is altered, the gas's volume must adjust to find a new equilibrium.

Pressure reflects how much force gas molecules exert on their container's walls. If external pressure decreases, such as in rising to higher altitudes, the gas pushes out more as it can "stretch" into more available space, thereby increasing its volume. Conversely, increasing the pressure through compression will decrease volume.

Temperature changes result in kinetic energy shifts for the molecules. Heating adds energy and encourages molecules to move fast and spread further apart, causing volume to increase if pressure remains constant. Cooling does the opposite, slowing molecules and pulling them closer.

• Lower pressure = Higher volume (if constant temperature)
• Higher temperature = Gas expands (with pressure constant)

Such relationships demonstrate the dynamic nature of gases and their connection to surrounding environmental changes. They are vital for predicting and controlling conditions in diverse scientific and industrial contexts.