Question

A volume of air is taken from the earth's surface, at \(19^{\circ} \mathrm{C}\) and \(1.00 \mathrm{~atm}\), to the stratosphere, where the temperature is \(-21^{\circ} \mathrm{C}\) and the pressure is \(1.00 \times 10^{-3}\) atm. By what factor is the volume increased?

Short answer

The volume increases by a factor of approximately 863.

Step-by-step solution

Convert Celsius to Kelvin

We need to convert the temperatures from Celsius to Kelvin. The formula to convert is: \( K = C + 273.15 \). \(19^{\circ} \mathrm{C} = 19 + 273.15 = 292.15 \mathrm{~K} \) and \(-21^{\circ} \mathrm{C} = -21 + 273.15 = 252.15 \mathrm{~K} \).

Apply the Combined Gas Law

The combined gas law states \( \frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2} \). We know \(P_1 = 1.00 \mathrm{~atm}\), \(T_1 = 292.15 \mathrm{~K}\), \(P_2 = 1.00 \times 10^{-3} \mathrm{~atm}\), and \(T_2 = 252.15 \mathrm{~K}\).

Rearrange the Formula to Solve for Volume Change

Rearrange the gas law to solve for the change in volume: \( \frac{V_2}{V_1} = \frac{P_1 \cdot T_2}{P_2 \cdot T_1} \).

Substitute the Known Values

Substitute the known values into the equation: \[ \frac{V_2}{V_1} = \frac{1.00 \times 252.15}{1.00 \times 10^{-3} \times 292.15} \].

Calculate the Volume Increase Factor

Calculate the result of the substitution: \( \frac{V_2}{V_1} = \frac{252.15}{0.29215} \approx 863 \).

Key concepts

Temperature Conversion

In situations involving gas laws, converting temperature from Celsius to Kelvin is crucial. Kelvin is the standard unit in gas law calculations because it avoids negative numbers, ensuring equations work correctly. The conversion is simple, achieved by adding 273.15 to the Celsius temperature. For example, converting from Celsius to Kelvin for the temperatures in our problem:

• At the Earth's surface: from \( 19^{\circ} \mathrm{C} \) to \( 19 + 273.15 = 292.15 \, \mathrm{K} \).
• In the stratosphere: from \( -21^{\circ} \mathrm{C} \) to \( -21 + 273.15 = 252.15 \, \mathrm{K} \).

Working in Kelvin ensures that relationships between temperature and other gas properties are proportional and accurate. This way, any changes in temperature affect the gas behavior directly and consistently.

Volume Change

Volume change of gases under varying conditions is directly related to pressure and temperature changes. This is expressed using the Combined Gas Law. The formula \( \frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2} \) allows us to calculate how much the volume of a gas changes when taken from one condition to another. In practice:

• Start by understanding the initial and final conditions: pressure \( P \), volume \( V \), and temperature \( T \).
• Use the formula rearranged as \( \frac{V_2}{V_1} = \frac{P_1 \cdot T_2}{P_2 \cdot T_1} \) to solve for the volume change factor.

This calculation helps in predicting how a gas will expand or compress when moved through different environments, like from the Earth's surface to the stratosphere.

Pressure

Pressure is a key factor in gas laws, affecting how gases behave under different conditions. Measured in atm, it represents force applied over a unit area. In the exercise, the transition from 1 atm on the surface to \(10^{-3}\) atm in the stratosphere shows a significant drop in pressure:

• Higher pressures compress gases, reducing volume.
• Lower pressures allow gases to expand, increasing volume.

Understanding pressure changes helps us predict how gases will behave when conditions change. It explains why airplanes and space shuttles are pressurized - to maintain a functional environment as they rise to higher, less dense altitudes.

Gas Laws

Gas laws provide the framework for understanding the behavior of gases under various conditions. The main laws include Boyle's Law, Charles's Law, and Gay-Lussac's Law. However, for complex changes, the Combined Gas Law is used as it incorporates:

• Boyle's principle that pressure and volume are inversely related when temperature is constant.
• Charles's principle that volume and temperature are directly related when pressure is constant.
• Gay-Lussac's principle that pressure and temperature are directly related when volume is constant.

In our exercise, the Combined Gas Law helps in understanding how a single packet of air will adapt to different temperatures and pressures as it moves to higher altitudes.

Stratosphere Conditions

The stratosphere, the layer of Earth’s atmosphere above the troposphere, has unique conditions. It starts about 10 km above the ground and extends up to 50 km. Conditions here affect how gases behave:

• Lower atmospheric pressure means gases expand more.
• Temperatures in the stratosphere rise with altitude, after an initial drop, due to ozone absorption of ultraviolet radiation.

In this exercise, moving air to the stratosphere means it experiences lower pressure and cooler temperatures initially. Predicting changes in volume under these conditions, using gas laws, showcases practical applications of theoretical knowledge in understanding atmospheric science and aerodynamics.