Exploring the Exciting World of Gas Expansion in a Hot-Air Balloon

How does the expansion of air in a hot-air balloon affect the number of moles of air present?

Given the initial volume, pressure, and temperature of the air in the balloon, along with the final volume after heating, what is the ratio of the number of moles of air in the heated balloon to the original number of moles of air in the balloon?

Answer

The ratio of the number of moles of air in the heated balloon to the original number of moles of air in the balloon is 1.47.

When a hot-air balloon is filled with air and heated, the expansion of the air inside the balloon leads to a change in the number of moles of air present. In this scenario, the balloon started with an initial volume of 3000 m³ at a temperature of 21°C and a pressure of 750 torr. As the air in the balloon is heated to 60°C, the volume of the air expands to 5000 m³.

To calculate the ratio of the number of moles of air in the heated balloon to the original number of moles of air in the balloon, we can use the combined gas equation with the assumption that the pressure remains constant due to openings in the balloon allowing air to flow in and out.

The combined gas equation when pressure is constant is:

[tex]\frac{V_1}{n_1T_1} = \frac{V_2}{n_2T_2}[/tex]

Where,

• [tex]n_1[/tex] = original number of moles of air in the balloon
• [tex]n_2[/tex] = number of moles of air in the heated balloon
• [tex]V_1[/tex] = initial volume of gas (3000 m³)
• [tex]V_2[/tex] = final volume of gas (5000 m³)
• [tex]T_1[/tex] = initial temperature of gas (21°C = 294K)
• [tex]T_2[/tex] = final temperature of gas (60°C = 333K)

By substituting the given values into the equation and solving for the ratio of [tex]\frac{n_2}{n_1}[/tex], we find that the ratio is 1.47. This means that there are 1.47 times more moles of air in the heated balloon compared to the original number of moles present in the balloon.