Question

A \(1.0-\mathrm{L}\) sample of air is collected at \(25^{\circ} \mathrm{C}\) at sea level \((1.00 \mathrm{atm}) .\) Estimate the volume this sample of air would have at an altitude of 15 \(\mathrm{km}\) ( see Fig.5.30) .At \(15 \mathrm{km},\) the pressure is about 0.1 \(\mathrm{atm} .\)

Short answer

The final volume of the air sample at an altitude of 15 km is 10 L, which can be calculated using the variant of the Ideal Gas Law: \(V2 = \frac{P1 × V1}{P2} = \frac{(1.0 \, L) × (1.00 \, atm)}{0.1 \, atm} = 10 \, L\).

Step-by-step solution

Write down the given values

Here, we are given: Initial volume of air, V1 = 1.0 L Initial pressure, P1 = 1.00 atm Temperature, T = 25°C = 25 + 273.15 = 298.15 K (convert Celsius to Kelvin) Final pressure at 15 km altitude, P2 = 0.1 atm We need to find the final volume, V2.

Use the variant of the Ideal Gas Law

As we know the initial and final pressure values and the initial volume, we can use the formula P1V1 = P2V2 to find the final volume.

Solve for V2

Rearrange the formula to find V2: \[V2 = \frac{P1 × V1}{P2}\] Now, substitute the given values and find the final volume: \[V2 = \frac{(1.0 \, L) × (1.00 \, atm)}{0.1 \, atm}\]

Calculate V2

Divide the nominator by the denominator to get the result: \[V2 = \frac{1.0}{0.1}\] \[V2 = 10 \, L\]

Final answer

The sample of air would have a volume of 10 L at an altitude of 15 km.

Key concepts

Gas Pressure

Gas pressure is a measure of the force that the gas exerts on the walls of its container. It's influenced by several factors, including temperature, volume, and the number of gas particles present. In general, gas pressure increases with more gas molecules in a given volume and decreases when the same number of molecules occupies a larger volume.
It's important to remember:

• Units: Pressure is typically measured in atmospheres (atm), but other units include pascals (Pa) and millimeters of mercury (mmHg).
• Instruments: Devices like barometers and manometers are commonly used to measure gas pressure.
• Physical Insights: High pressure means particles are moving rapidly and colliding more frequently with the container walls.

The ideal gas law, represented as PV=nRT, is particularly useful in understanding how pressure changes in different scenarios. Here, P stands for pressure, V for volume, n for the number of moles, R for the ideal gas constant, and T for temperature. Using this equation helps in relating pressure to other factors like volume and temperature.

Volume and Pressure Relationship

One of the key principles in gas behavior is the volume and pressure relationship, which is described by Boyle's Law. This law states that the pressure of a given mass of gas is inversely proportional to its volume, as long as the temperature remains constant.
Mathematically speaking:

• The equation is expressed as \(P_1V_1 = P_2V_2\), where \(P_1\) and \(V_1\) are the initial pressure and volume, and \(P_2\) and \(V_2\) are the final pressure and volume.
• If the volume of a gas decreases, its pressure increases and vice versa. This is why squeezing a balloon makes it more likely to pop.
• Illustrative Thought: Imagine gas molecules in a shrinking space; as the space shrinks, they collide with the walls more frequently, increasing the pressure.

Boyle’s Law is a special case within the ideal gas law where the temperature and the amount of gas do not change. This relates directly to our problem since the temperature remains constant while the volume changes due to a change in pressure at higher altitudes.

Altitude Effects on Gases

As altitude increases, the pressure exerted by the atmosphere decreases. This happens because there is less atmospheric gas the higher up you go. At sea level, the atmospheric pressure is about 1 atm, while at high altitudes, like 15 kilometers above sea level, it drops significantly, here to about 0.1 atm.
When considering altitude:

• The decrease in pressure impacts the volume of gases if their temperature and mass remain constant.
• The relationship can be described using the Boyle’s Law formula. This explains why the 1-L sample of air expands to 10 L when the pressure drops from 1 atm to 0.1 atm at 15 km altitude.
• Practical Implications: This concept affects aviation as the lower pressure at high altitudes necessitates pressurizing aircraft cabins to ensure passenger comfort and safety.

Understanding altitude effects on gases is crucial not only for flying but also for climbing mountains and other activities involving significant elevation changes, where human bodies must adjust to lower pressure and oxygen levels.