The total mass that can be lifted by a balloon is given by the difference between the mass of air displaced by the balloon and the mass of the gas inside the balloon. Consider a hot air balloon that approximates a sphere 5.00 m in diameter and contains air heated to 65 C. The surrounding air temperature is 21 C. The pressure inside the balloon is equal to the atmospheric pressure, which is 745 torr.
a. What total mass can the balloon lift? Assume that the average molar mass of air is 29.0 g/mol. (Hint: heated air is less dense than cool air
b. If the ballon is filled with enough helium at 21 C and 745 torr to achieve the same volume as in part a, what total mass can the balloon lift?
c. What mass oculd the hot air balloon in part a lift if it were on the ground in Denver, Colorado, where a typical atmospheric pressure is 630 torr?
To find the total mass that can be lifted by the balloon, we need to consider the difference between the mass of air displaced by the balloon and the mass of gas inside the balloon.
a. First, let's calculate the mass of air displaced by the balloon. We can use the ideal gas law equation for this:
PV = nRT
Where:
P = Pressure
V = Volume
n = number of moles
R = Ideal gas constant (0.0821 L·atm/(mol·K))
T = Temperature (convert to Kelvin)
Given:
Diameter of balloon = 5.00 m
Radius of balloon (r) = 2.50 m
Temperature inside the balloon (T1) = 65 °C = 338 K
Temperature of surrounding air (T2) = 21 °C = 294 K
Pressure inside the balloon (P) = 745 torr
Volume of a sphere = (4/3)πr^3
Now, we can calculate the volume of the balloon:
V = (4/3)πr^3
= (4/3) * 3.14159 * (2.50)^3
= 65.44956 m^3
Next, let's calculate the mass of air displaced by the balloon at the given temperature and pressure:
Using the ideal gas law equation PV = nRT, we can solve for n:
n = PV / RT
Now, we need to calculate the number of moles (n) of air:
n = (745 torr * 65.44956 m^3) / (0.0821 L·atm/(mol·K) * 338 K)
= 18.69 moles
To calculate the mass of air displaced, we multiply the number of moles by the molar mass of air:
Mass of air displaced = n * Molar mass of air
= 18.69 moles * 29.0 g/mol
= 541.8 g
Now, let's calculate the mass of the gas inside the balloon:
Mass of gas inside = n * Molar mass of gas
Since the balloon is filled with air, the molar mass of the gas is the same as the molar mass of air:
Mass of gas inside = 18.69 moles * 29.0 g/mol
= 541.8 g
Finally, we can calculate the total mass that the balloon can lift:
Total mass lifted = Mass of air displaced - Mass of gas inside
= 541.8 g - 541.8 g
= 0 g
Therefore, the hot air balloon in part a cannot lift any additional mass.
b. If the balloon is filled with helium, we can repeat the calculations using the molar mass of helium. The molar mass of helium is approximately 4.0 g/mol.
The volume of the balloon (V) remains the same.
Number of moles (n) = PV / RT
= (745 torr * 65.44956 m^3) / (0.0821 L·atm/(mol·K) * 338 K)
= 18.69 moles
The mass of gas inside = n * Molar mass of gas
= 18.69 moles * 4.0 g/mol
= 74.76 g
Total mass lifted = Mass of air displaced - Mass of gas inside
= 541.8 g - 74.76 g
= 467.04 g
Therefore, the hot air balloon in part b can lift a total mass of 467.04 g.
c. To calculate the mass the hot air balloon in part a can lift on the ground in Denver, Colorado, we need to change the atmospheric pressure.
The new atmospheric pressure (P) is 630 torr.
Recalculate the number of moles (n) using the new pressure:
n = (P * V) / (R * T)
= (630 torr * 65.44956 m^3) / (0.0821 L·atm/(mol·K) * 338 K)
= 16.91 moles
Calculate the mass of air displaced:
Mass of air displaced = n * Molar mass of air
= 16.91 moles * 29.0 g/mol
= 490.39 g
Total mass lifted = Mass of air displaced - Mass of gas inside
= 490.39 g - 541.8 g
= -51.41 g
Therefore, the hot air balloon in part a cannot lift any additional mass when on the ground in Denver, Colorado.
a. To determine the total mass that the hot air balloon can lift, we need to calculate the difference between the mass of air displaced and the mass of the hot air inside the balloon.
Step 1: Calculate the volume of the balloon.
Given that the diameter of the balloon is 5.00 m, we can calculate the radius by dividing it by 2:
Radius = 5.00 m / 2 = 2.50 m
The volume of a sphere can be calculated using the formula:
Volume = (4/3) * π * (radius^3)
Volume = (4/3) * 3.14 * (2.50 m)^3
Volume ≈ 65.45 m^3
Step 2: Calculate the mass of air displaced by the balloon.
Air is less dense when heated, so we need to take into account the difference in density between the heated air inside the balloon and the cooler air outside. To do this, we can use the ideal gas law.
The ideal gas law states: PV = nRT
Where:
P = pressure (in torr)
V = volume (in m^3)
n = moles of gas
R = ideal gas constant (0.0821 L * atm / K * mol or 62.36 L * torr / K * mol)
T = temperature (in Kelvin)
First, we need to convert the temperature from Celsius to Kelvin:
T_hot_air = 65 C + 273.15 = 338.15 K
T_surrounding_air = 21 C + 273.15 = 294.15 K
Next, we need to calculate the number of moles of air inside the balloon:
n = (P * V) / (R * T)
n = (745 torr * 65.45 m^3) / (62.36 L * torr / K * mol * 338.15 K)
n ≈ 40.75 mol
The average molar mass of air is given as 29.0 g/mol, so we can calculate the mass of air displaced:
Mass_displaced = n * molar_mass
Mass_displaced = 40.75 mol * 29.0 g/mol
Mass_displaced ≈ 1182.25 g
Step 3: Calculate the mass of the hot air inside the balloon.
The mass can be calculated using the density formula:
Density = mass / volume
Rearranging the formula, we have:
Mass = density * volume
Density_hot_air = Mass_hot_air / Volume
Density_surrounding_air = Mass_surrounding_air / Volume
Density_hot_air = (Mass_hot_air + Mass_displaced) / Volume
Density_surrounding_air = Mass_surrounding_air / Volume
Since the density of hot air is less than the density of surrounding air, the hot air will rise. Therefore, we can assume that the density of hot air is equal to the density of surrounding air.
Mass_hot_air + Mass_displaced = Mass_surrounding_air
Finally, substituting in the known values:
Mass_hot_air = Mass_surrounding_air - Mass_displaced
Mass_hot_air = density_surrounding_air * volume - Mass_displaced
Using the ideal gas law, we can calculate the density of the surrounding air:
density_surrounding_air = (P_surrounding_air * molar_mass) / (R * T_surrounding_air)
density_surrounding_air = (745 torr * 29.0 g/mol) / (62.36 L * torr / K * mol * 294.15 K)
Now we can calculate the mass of hot air:
Mass_hot_air = (density_surrounding_air * volume) - Mass_displaced
Substitute in the known values:
Mass_hot_air ≈ (0.9165 g/L * 65.45 m^3) - 1182.25 g
Mass_hot_air ≈ 60,044.9 g - 1182.25 g
Mass_hot_air ≈ 58,862.65 g
Step 4: Calculate the total mass the balloon can lift.
The total mass lifted by the balloon is the difference between the mass of air displaced and the mass of hot air inside the balloon:
Total_mass_lifted = Mass_displaced - Mass_hot_air
Total_mass_lifted ≈ 1182.25 g - 58,862.65 g
Total_mass_lifted ≈ -57,680.4 g
The negative sign indicates that the balloon is not able to lift any mass, as the mass of the hot air inside is greater than the mass of the air displaced.