Exciting Calculation of Air Moles in Hot-Air Balloon Scenario

How 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?

Given data:

Original volume of air in balloon: 4.00 x 10^3 m3

Pressure: 745 torr

Initial temperature: 218°C

Heated temperature: 62°C

Final volume of air in balloon: 4.20 x 10^3 m3

Calculation of the Ratio of Moles of Air in Heated Balloon to Original Moles

To find the ratio of moles of air in the heated balloon to the original moles, we need to calculate the number of moles in both scenarios.

Let's start by calculating the original number of moles of air in the balloon using the ideal gas equation: PV = nRT.

Given:

Pressure (P) = 745 torr = 99308.5 Pa

Volume (V) = 4.00 x 10^3 m3

Temperature (T) = 218°C = 491K

Gas constant (R) = 8.314 Pa.m3/mol.K

After calculations, the original number of moles of air in the balloon is determined to be 97309.4 mol.

Next, we calculate the number of moles of air in the heated balloon using the same ideal gas equation.

Given:

Pressure (P) = 745 torr = 99308.5 Pa

Volume (V) = 4.20 x 10^3 m3

Temperature (T) = 626°C = 899K

Gas constant (R) = 8.314 Pa.m3/mol.K

Upon computation, the number of moles of air in the heated balloon is calculated to be 55804.1 mol.

Finally, we determine the ratio of moles of air in the heated balloon to the original moles by dividing the moles in the heated balloon by the original moles.

Ratio = 55804.1 mol / 97309.4 mol ≈ 0.5735

Therefore, the exciting calculation reveals that 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 approximately 0.5735.