A hot air balloon has a volume of 5.00x10^6 liters. Assuming a density of air of 1.293g/L at 25C and 1atm pressure, what temperature must the operator of the balloon achieve in order for the balloon to lift a 500.0kg payload?
How do I set this up???????
To set up this problem, you can use the ideal gas law equation, which states:
PV = nRT
Where:
P = pressure
V = volume
n = number of moles
R = ideal gas constant
T = temperature
Since you are given the volume of the hot air balloon (5.00x10^6 liters) and the pressure (1 atm), you need to find the temperature required to lift the payload. Here's how to set it up:
1. Convert the volume of the hot air balloon to m³:
5.00x10^6 liters * (1 m³ / 1000 liters) = 5000 m³
2. Calculate the number of moles of air:
To calculate the number of moles, we need to know the mass of air in grams. Given the density of air at 25°C (1.293 g/L), we can use the formula:
Density = Mass / Volume
Rearranging the equation, we find that Mass = Density * Volume.
Mass = 1.293 g/L * 5.00x10^6 L = 6.465x10^6 g
Now, convert the mass from grams to moles:
Moles = Mass / Molar mass of air
The molar mass of air is approximately 28.97 g/mol.
Moles = 6.465x10^6 g / 28.97 g/mol ≈ 223,370 mol
3. Plug in the values into the ideal gas law equation:
P * V = n * R * T
We know:
P = 1 atm
V = 5000 m³
n = 223,370 mol
R = ideal gas constant = 0.0821 L·atm/(mol·K)
Rearranging the equation to solve for T:
T = (P * V) / (n * R)
Therefore:
T = (1 atm * 5000 m³) / (223,370 mol * 0.0821 L·atm/(mol·K))