Some say that the hot air balloons with which the Montgolfier brothers performed their first flight had a volume of 1700 cubic metres and could lift 780 kilograms (that includes the balloon, basket and payload). Assume that the balloon took off on a day when the sea level pressure was 1013.25 hPa.
Given an outside temperature of 10.0 degrees Celsius, compute the temperature of the air (in degrees Celsius) in the balloon.
Given the outside air temperature (which is 283.15 K), we can compute the air density:
P=p/RT=101325/287.00*283.15 =1.24686kg/m^3
This gives us-
L=mg=pVg (delta T/T + delta T)
Plugging in the numbers this gives us 780*9.81 (Delta T/T+Delta T) = 0.368
Using this ratio and t=283.15K, we find that Delta T= 165 K, so T+Delta T= 175 degrees Celsius
Hope this helped!
To compute the temperature of the air inside the balloon, we need to use the ideal gas law, which states that PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin.
First, we need to convert the temperature of 10.0 degrees Celsius to Kelvin. To do this, we can use the equation T(K) = T(C) + 273.15, where T(K) is the temperature in Kelvin and T(C) is the temperature in Celsius.
So, the temperature of 10.0 degrees Celsius in Kelvin is:
T(K) = 10.0 + 273.15 = 283.15 K
Now, let's rearrange the ideal gas law equation to solve for temperature:
T = PV / (nR)
The volume of the balloon is given as 1700 cubic meters, and the pressure is 1013.25 hPa (hectopascals), which is equivalent to 1013.25 * 100 Pa. However, we also need to convert the pressure to Pascals, so the pressure in Pascals is:
P = 1013.25 hPa * 100 Pa/hPa = 101,325 Pa
The ideal gas constant, R, is 8.314 J/(mol·K).
To calculate the number of moles, we need to know the molar mass. However, since the composition of the air is not given, we can assume an average molar mass of air, which is approximately 28.97 g/mol.
We have the weight of the balloon, basket, and payload, which is 780 kg. To convert this to grams, we multiply by 1000, so the weight in grams is:
m = 780 kg * 1000 g/kg = 780,000 g
To find the number of moles, we divide the mass by the molar mass:
n = m / M
Plugging in the values, we get:
n = 780,000 g / 28.97 g/mol ≈ 26,901.78 mol
Now, we can substitute the values into the equation:
T = (101,325 Pa * 1700 m^3) / (26,901.78 mol * 8.314 J/(mol·K))
Simplifying the equation gives us:
T ≈ 791 K
Finally, we can convert the temperature back to degrees Celsius by subtracting 273.15:
T(C) ≈ 791 - 273.15 ≈ 517.85 degrees Celsius
Therefore, the temperature of the air inside the balloon is approximately 517.85 degrees Celsius.