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?
a. Well, let me calculate that for you, but first, let me tell you, this balloon is really hot stuff! Okay, let's get on with the calculation. To find the total mass the hot air balloon can lift, we need to calculate the difference between the mass of air displaced by the balloon and the mass of the hot air inside.
First, we need to find the mass of air displaced by the balloon. Since the balloon is a sphere, we can use the formula for the volume of a sphere (V = 4/3 * π * r^3), where r is the radius (diameter/2). So, the volume of the balloon is V = 4/3 * π * (5.00/2)^3.
Now, to calculate the mass of air displaced, we use the density of air (ρ = mass/volume). The density of air can be calculated using the ideal gas law (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. We can rearrange the equation to get the density (ρ = P/(RT)).
Using the given pressure inside the balloon and the average molar mass of air, we can calculate the density of air at 65 °C. Then, we multiply the density by the volume of the balloon to find the mass of air displaced.
Next, we need to find the mass of the hot air inside the balloon. Again, we use the ideal gas law, but this time with the given temperature and pressure inside the balloon. This gives us the number of moles of air, which we can then multiply by the average molar mass of air to find the mass.
Finally, we subtract the mass of the hot air from the mass of the air displaced to find the total mass the balloon can lift. Phew, that was a lot of math! But don't worry, I'll do it for you. Give me a moment.
b. Ah, helium, the lighter-than-air gas. It's like the celebrity of gases, always floating above the rest. If we fill the balloon with helium at the same volume and temperature, things will be a lot lighter. Helium has a much lower molar mass than air, so the mass of the helium inside the balloon will be much smaller. Let me calculate it for you.
c. Ah, the Denver factor. The mile-high city, where the air is thinner and the balloons might need a little extra help. If we consider the lower atmospheric pressure in Denver, the mass the balloon can lift will be different. We'll need to recalculate using the given pressure in Denver instead of the standard atmospheric pressure. Let me figure it out for you.
Alright, get ready for some calculations, and hold on tight!
To solve these problems, we will be using the ideal gas law equation. The ideal gas law is represented as:
PV = nRT
where:
P is the pressure
V is the volume
n is the number of moles
R is the ideal gas constant (0.0821 L·atm/(mol·K))
T is the temperature in Kelvin
Let's solve these problems step-by-step.
a. What total mass can the balloon lift?
1. First, let's calculate the volume of the sphere using the given diameter.
The diameter of the balloon is 5.00 m, so the radius is half of that, which is 2.50 m.
The formula for the volume of a sphere is V = (4/3)πr^3.
So, the volume of the balloon is V = (4/3) * π * (2.50)^3 = 65.44 m^3.
2. Next, let's calculate the mass of air displaced by the balloon.
To calculate the mass of the air displaced, we need to find the difference in density between the inside and outside of the balloon.
The density of air can be approximated using the ideal gas law: PV = nRT.
First, convert the temperatures to Kelvin:
Inside temperature = 65°C + 273.15 = 338.15 K
Outside temperature = 21°C + 273.15 = 294.15 K
Now, let's calculate the densities:
Density inside the balloon = (P inside * Molar mass of air) / (R * T inside)
Density outside the balloon = (P outside * Molar mass of air) / (R * T outside)
Using the given information:
P inside = atmospheric pressure = 745 torr = 745/760 atm
P outside = atmospheric pressure = 1 atm
Molar mass of air = 29.0 g/mol
R = 0.0821 L·atm/(mol·K)
T inside = 338.15 K
T outside = 294.15 K
Now we can calculate the densities:
Density inside the balloon = (745/760 * 29.0) / (0.0821 * 338.15)
Density outside the balloon = (1 * 29.0) / (0.0821 * 294.15)
3. Finally, calculate the mass of air displaced:
Mass of air displaced = difference in density * volume of the balloon
Mass of air displaced = (Density outside the balloon - Density inside the balloon) * Volume
Mass of air displaced = (Density outside - Density inside) * Volume
Mass of air displaced = (Density outside) * Volume - (Density inside) * Volume
Substituting the values we calculated, we get:
Mass of air displaced = (Density outside - Density inside) * Volume
Mass of air displaced = (1.18 kg/m^3 - 1.05 kg/m^3) * 65.44 m^3
b. If the balloon 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?
To solve this part, we need to calculate the mass of helium using the same steps as in part a, but using the molar mass of helium.
c. What mass could 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 solve this part, we need to use the same steps as in part a, but with the given atmospheric pressure of 630 torr.
To determine the total mass that a balloon can lift, we need to consider the difference in the mass of air displaced by the balloon and the mass of the gas inside the balloon. Let's break down each part of the question:
a. To find the total mass that the hot air balloon can lift when filled with hot air, we need to calculate the mass of air displaced by the balloon and subtract the mass of the gas inside the balloon.
1. Calculate the volume of the balloon:
The volume of a sphere can be calculated using the formula: V = (4/3) * π * r^3, where r is the radius of the sphere. In this case, the diameter is given as 5.00 m, so the radius is half of the diameter (2.50 m). Therefore, the volume is:
V = (4/3) * π * (2.50 m)^3.
2. Calculate the mass of air displaced:
To find the mass of air displaced, we can use the ideal gas law formula: PV = nRT, where P is the pressure, V is the volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature in Kelvin.
First, convert the temperature to Kelvin by adding 273.15:
Temperature in Kelvin = 65 C + 273.15.
Next, calculate the number of moles of air:
n = PV / RT, where P is the pressure, V is the volume, R is the ideal gas constant, and T is the temperature.
3. Calculate the mass of air displaced:
Mass of air displaced = n * Average molar mass of air.
4. Calculate the mass of the gas inside the balloon:
The mass of the gas inside the balloon is determined by multiplying the molar mass (given in grams/mol) by the number of moles.
5. Subtract the mass of the gas inside the balloon from the mass of air displaced:
Total mass lifted = Mass of air displaced - Mass of gas inside the balloon.
b. To find the total mass that the balloon can lift when filled with helium, we repeat the same steps as in part a, but with different values for the molar mass of helium.
1. Calculate the volume of the balloon (same as in part a).
2. Calculate the number of moles of helium using the ideal gas law (same as in part a).
3. Calculate the mass of helium displaced using the molar mass of helium.
4. Subtract the mass of the gas inside the balloon (helium) from the mass of helium displaced to find the total mass lifted.
c. To find the mass that the hot air balloon can lift in Denver, Colorado, with a different atmospheric pressure, we can use the same steps as in part a, but with the new pressure value.
1. Calculate the volume of the balloon (same as in part a).
2. Calculate the number of moles of air using the ideal gas law (same as in part a), but with the new pressure value.
3. Calculate the mass of air displaced using the molar mass of air.
4. Subtract the mass of the gas inside the balloon (air) from the mass of air displaced to find the total mass lifted.
By following these steps and plugging in the given values, you can calculate the total mass that can be lifted by the balloon in each scenario.