The air inside a hot air balloon has an average temperature of 71.7° C. The outside air has a temperature of 11.1° C. What is the ratio of the density of air in the balloon to the density of air in the surrounding atmosphere?
Since the pressure is very nearly the same inside and outside the balloon, density is inversely proportional to absolute temperature.
rho_inside/rho_outside = T_outside/T_inside
= 284.3/344.9 = 0.824
To find the ratio of the density of air in the balloon to the density of air in the surrounding atmosphere, we can make use of the ideal gas law. According to the ideal gas law, the density of a gas is directly proportional to its pressure and inversely proportional to its temperature.
Let's assume that the pressure inside the balloon and the pressure outside the balloon are equal. This is a reasonable assumption as the balloon is at equilibrium with the surrounding atmosphere.
The ratio of the densities can be calculated as follows:
ρ_balloon/ρ_atmosphere = (P_balloon * M) / (P_atmosphere * M)
where,
ρ_balloon is the density of air in the balloon
ρ_atmosphere is the density of air in the surrounding atmosphere
P_balloon is the pressure inside the balloon
P_atmosphere is the pressure outside the balloon
M is the molar mass of air
Since the pressure inside the balloon is equal to the pressure outside the balloon, we can simplify the expression:
ρ_balloon/ρ_atmosphere = M/M = 1
Therefore, the ratio of the density of air in the balloon to the density of air in the surrounding atmosphere is 1.
To find the ratio of the density of air in the balloon to the density of air in the surrounding atmosphere, we need to use the ideal gas law. The ideal gas law states that the pressure (P), volume (V), and temperature (T) of a gas are related by the formula PV = nRT, where n is the number of moles of gas and R is the ideal gas constant.
Since we are only interested in the ratio of densities, we can assume that the volume of air in the balloon and the surrounding atmosphere are equal. Therefore, we can simplify the equation to P1/T1 = P2/T2, where P1 and P2 are the pressures and T1 and T2 are the temperatures of the air inside the balloon and the surrounding atmosphere, respectively.
First, let's convert the temperatures from Celsius to Kelvin:
T1 = 71.7 + 273.15 = 344.85 K
T2 = 11.1 + 273.15 = 284.25 K
Now, plug in the values into the equation and solve for the pressure ratio:
P1/T1 = P2/T2
P1 = P2 * (T1/T2)
P1/P2 = T1/T2
P1/P2 = 344.85/284.25
P1/P2 = 1.210
Therefore, the ratio of the density of air in the balloon to the density of air in the surrounding atmosphere is approximately 1.210.