Question

When J.F. Piccard made a stratosphere flight in a balloon, the balloon seemed to be only half filled as it left the ground near Detroit. The gas temperature was about \(27^{\circ} \mathrm{C}\), the pressure \(700 \mathrm{~mm}\), and the volume of gas in the balloon 80,000 cubic feet. What was the gas volume at high altitude where the temperature was \(-3^{\circ} \mathrm{C}\), and the pressure \(400 \mathrm{~mm}\) ?

Short answer

The gas volume at high altitude is approximately \(103,597.30\mathrm{ft}^3\).

Step-by-step solution

Understand the given information

We are given the initial state of the gas in the balloon: - Initial temperature, \(T_1 = 27^{\circ}\mathrm{C}\) - Initial pressure, \(P_1 = 700\mathrm{mm}\) - Initial volume, \(V_1 = 80,000\mathrm{ft}^3\) We are also given the final state of the gas in the balloon: - Final temperature, \(T_2 = -3^{\circ}\mathrm{C}\) - Final pressure, \(P_2 = 400\mathrm{mm}\) Our task is to find the final volume, \(V_2\). To use the ideal gas law, we need to convert the temperature to Kelvin and the pressure to atmosphere (atm).

Convert temperatures to Kelvin

To convert temperatures from Celsius to Kelvin, we add 273.15 to the temperature in Celsius: - Initial temperature in Kelvin, \(T_1 = 27 + 273.15 = 300.15\mathrm{K}\) - Final temperature in Kelvin, \(T_2 = -3 + 273.15 = 270.15\mathrm{K}\)

Convert pressures to atmosphere

To convert pressures from mmHg to atm, we can use the conversion: \(1\mathrm{atm} = 760\mathrm{mmHg}\) - Initial pressure in atm, \(P_1 = \frac{700}{760} = 0.921\mathrm{atm}\) - Final pressure in atm, \(P_2 = \frac{400}{760} = 0.526\mathrm{atm}\)

Apply the ideal gas law

Since the amount of gas in the balloon is constant, we can use the ideal gas law formula: \(\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\) Now we can plug in the known values: \(\frac{0.921 \times 80,000}{300.15} = \frac{0.526 \times V_2}{270.15}\)

Solve for the final volume

Now we can solve for \(V_2\): \(V_2 = \frac{0.526 \times 80,000 \times 270.15}{0.921 \times 300.15} \approx 103,597.30\mathrm{ft}^3\) So the gas volume at high altitude is approximately \(103,597.30\mathrm{ft}^3\).

Key concepts

Temperature Conversion

Temperature conversion is essential when dealing with gas laws because most calculations are performed using the Kelvin scale. The Kelvin scale is an absolute temperature scale starting at absolute zero, making it ideal for scientific calculations.
- To convert from Celsius to Kelvin, simply add 273.15 to the Celsius value. For example, converting an initial temperature of \(27^{\circ} \mathrm{C}\) gives \(27 + 273.15 = 300.15 \mathrm{K}\).
- Similarly, a temperature of \(-3^{\circ} \mathrm{C}\) would be \(-3 + 273.15 = 270.15 \mathrm{K}\).
Converting temperatures ensures that the calculations remain consistent and accurate, as gas volumes can vary significantly with temperature changes. Always remember to convert any Celsius temperatures to Kelvin when using formulas involving the ideal gas law.

Pressure Conversion

Pressure needs to be converted into consistent units when applying the ideal gas law. In chemistry and physics, it is standard to use atmospheres (atm) as the unit for pressure.
- One atmosphere is equivalent to 760 mmHg. To convert mmHg to atm, divide the pressure in mmHg by 760. For example, a pressure of \(700\mathrm{mmHg}\) would be \(\frac{700}{760} \approx 0.921\mathrm{atm}\).
- Similarly, \(400\mathrm{mmHg}\) converts to \(\frac{400}{760} \approx 0.526\mathrm{atm}\).
Converting pressures is crucial to ensure consistency with temperature (in Kelvin) and volume (in cubic units) when using the ideal gas law for calculations involving the change in gas volume.

Gas Volume Calculation

Gas volume calculation involves understanding how the volume of gas changes with temperature and pressure. Using the ideal gas law equation, you can determine the new volume of a gas when its temperature and pressure change.
- The ideal gas law formula is \(\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\). Here, \(P\) is pressure, \(V\) is volume, and \(T\) is temperature.
- To find the final volume \(V_2\), reorganize the formula: \(V_2 = \frac{P_1 V_1 T_2}{P_2 T_1}\).
- Plug the known values: \(V_2 = \frac{0.921 \times 80,000 \times 270.15}{0.526 \times 300.15} \approx 103,597.30 \mathrm{ft}^3\).
Calculating gas volume changes allows us to predict how a gas behaves under different environmental conditions, such as those experienced during a stratosphere flight.

Boyle's and Charles's Laws

Boyle's and Charles's Laws are specific cases of the ideal gas law.
- **Boyle’s Law** states that the volume of a gas is inversely proportional to its pressure when the temperature is constant, described mathematically as \(P_1 V_1 = P_2 V_2\).
- **Charles’s Law** indicates that the volume of a gas is directly proportional to its temperature when pressure is constant, represented by \(\frac{V_1}{T_1} = \frac{V_2}{T_2}\).
These laws simplify the Ideal Gas Law equation for particular conditions where either pressure or temperature remains constant. They help us understand how a single variable affects gas behavior, making predictions about the gas’s volume changes in more complex scenarios, such as during high-altitude flights where both temperature and pressure vary.