A typical volume of a modern hot air balloon is 2500 cubic metres, and a typical maximum temperature of the hot air is 120 degrees Celsius. Given these figures, and an outside air temperature and density of 15 \(^{\circ}C\) and 1.225 \(kg/m^3\) respectively, compute the maximum mass (in kilograms) of the balloon, basket and payload.
To compute the maximum mass of the balloon, basket, and payload, we need to consider the ideal gas law, which states:
PV = nRT
where:
P is the pressure (which is the same inside and outside the balloon)
V is the volume of the balloon
n is the number of moles of gas
R is the ideal gas constant
T is the temperature
Rearranging the formula to solve for n, we get:
n = PV / RT
Since the pressure is the same inside and outside the balloon, we can cancel it out. Thus, the formula becomes:
n = V / RT
To find the number of moles, we need to convert the temperature from degrees Celsius to Kelvin. The Kelvin temperature scale can be obtained by adding 273.15 to the Celsius temperature.
Given:
Volume of the balloon (V) = 2500 cubic meters
Temperature of the hot air (T) = 120 degrees Celsius = 120 + 273.15 = 393.15 Kelvin
Ideal gas constant (R) = 8.3145 J/(mol·K) (this value is derived from the ideal gas equation)
Now, let's calculate the number of moles using the given formula:
n = V / RT
n = 2500 / (8.3145 * 393.15)
Calculating the value for n, we get:
n ≈ 0.7749 moles
To find the maximum mass, we need to consider the molar mass of air (which consists mainly of nitrogen and oxygen) and multiply it by the number of moles.
The molar mass of air is approximately 28.97 g/mol.
So, the mass of the balloon, basket, and payload is:
mass = n * molar mass
mass = 0.7749 * 28.97
Calculating the value for mass:
mass ≈ 22.4185 grams
Finally, we need to convert the mass from grams to kilograms:
max mass = 22.4185 grams = 0.0224185 kg
Therefore, the maximum mass of the balloon, basket, and payload is approximately 0.0224185 kilograms.
To compute the maximum mass of the balloon, basket, and payload, we need to consider the buoyant force and the weight of the system.
The buoyant force is the force that opposes the weight of an object submerged in a fluid (in this case, air). It is equal to the weight of the displaced air. The weight of the system is the sum of the weights of the balloon, basket, and payload.
First, let's calculate the displaced air mass by subtracting the volume of the balloon from the total volume of air displaced:
Displaced air volume = Total volume of air - Volume of balloon
Since the total volume of air is unknown, we cannot directly calculate the displaced air mass. However, we can use the ideal gas law to approximate it. The ideal gas law states that:
PV = nRT
Where:
P = pressure (assumed constant)
V = volume
n = number of moles of gas
R = gas constant (8.314 J/(K.mol))
T = temperature (in Kelvin)
To calculate the number of moles (n) of air in the balloon, we need to convert the temperature from Celsius to Kelvin:
T(Kelvin) = T(Celsius) + 273.15
Temperature of the hot air = 120 + 273.15 = 393.15 K
Temperature of the outside air = 15 + 273.15 = 288.15 K
Now, let's calculate the moles of air using the ideal gas law:
n = (PV) / (RT)
Since the pressure (P), volume (V), and temperature (T) are known, we can substitute these values into the equation to find the moles of air.
Next, we need to calculate the displaced air mass by multiplying the moles of air by the molar mass of air. The molar mass of air is approximately 28.97 g/mol.
Displaced air mass = Moles of air x Molar mass of air
Now, to calculate the weight of the system, we need to consider the density of the outside air. The density (ρ) of air is given as 1.225 kg/m^3.
Weight of the system = Displaced air mass x Density of air
Finally, we have the weight of the system. To find the maximum mass of the balloon, basket, and payload, we need to subtract the weight of the balloon itself (since it is filled with hot air and provides lift) from the weight of the system.
Maximum mass of balloon, basket, and payload = Weight of the system - Weight of the balloon
Please substitute the values provided and use the equations mentioned above to calculate the maximum mass of the balloon, basket, and payload.