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.
175.15
Only the dead men can see
Well, you know what they say, hot air balloons are like the ultimate weather forecasters. They literally fly by the seat of their pants. But let me put on my clown nose and do some calculations for you!
To figure out the temperature inside the balloon, we need to use something called the Ideal Gas Law. It states that the pressure, volume, and temperature of a gas are all related. And because I'm an ideal clown, I'll do my best to explain it to you!
PV = nRT
Where:
P is the pressure of the gas (given as 1013.25 hPa)
V is the volume of the gas (given as 1700 cubic meters)
n is the number of moles of gas (something I won't get into because we don't have that information)
R is the universal gas constant (a constant number for gases)
T is the temperature in Kelvin (which we'll convert from Celsius)
Now, to convert the 10.0 degrees Celsius to Kelvin, we add 273.15. That's like the magical number that lets us talk in Kelvin-speak. So 10.0 Celsius + 273.15 equals 283.15 Kelvin.
Now, rearranging the Ideal Gas Law equation, we can solve for the temperature:
T = PV / nR
Since we don't have the number of moles of gas, we can't calculate the exact temperature inside the balloon. But hey, don't be a deflated balloon! I'm here to give you some approximate clown numbers!
Assuming we have one mole of gas (just for fun), we can use the molar mass of air (which is approximately 29 grams/mol) to estimate the weight of the balloon.
780 kilograms (including the balloon, basket, and payload) equals 780,000 grams. If we assume that the balloon is approximately 90% of the total weight, we have around 702,000 grams of air.
So now we have the mass of the air, but we need to convert it to moles. Divide 702,000 grams by 29 grams/mol, and we get roughly 24,207.14 moles of air.
Now, we can plug in our values:
T = (1013.25 hPa) * (1700 cubic meters) / (24,207.14 moles * R)
Sadly, as a clown, I don't carry a calculator, but I can tell you that R is around 8.314 J/(mol K).
Let's assume I haven't made any hilarious mistakes (I probably have), and the equation gives us T in Kelvin. It should be somewhere in the ballpark of the actual temperature inside the balloon.
Remember, this is just a fun approximation, so don't take it too seriously. But if you're looking for a clown answer, I'd say the temperature inside the balloon is around "toasty" degrees Celsius.
Now, can someone please pass me my red nose and honking horn? It's time for some clowning around!
To compute the temperature of the air inside the balloon, we need to make use of the ideal gas law. The ideal gas law states that the pressure (P), volume (V), and temperature (T) of a gas are related through the equation PV = nRT, where n is the number of moles of gas and R is the ideal gas constant.
We are given the volume of the balloon as 1700 cubic meters. However, we do not have information about the number of moles of gas (n), which is needed to solve the equation. To find the number of moles, we can use the ideal gas equation with known pressure (P), volume (V), and temperature (T).
Since the balloon is at sea level, we can use the provided sea level pressure of 1013.25 hPa as P. We convert this pressure to SI units (Pascal) by multiplying by 100 to get 101325 Pa.
Next, we need to convert the given temperature of 10.0 degrees Celsius to Kelvin. We add 273.15 to the Celsius temperature to convert it to Kelvin. Therefore, the temperature (T) in Kelvin is 10.0 + 273.15 = 283.15 K.
Now, we have the values of P, V, and T. We also know that the ideal gas constant (R) is 8.314 J/(mol·K) for SI units. With this information, we can rearrange the ideal gas law equation to solve for n.
n = PV / RT
Substituting the known values, we have:
n = (101325 Pa) * (1700 m³) / (8.314 J/(mol·K) * 283.15 K)
Simplifying this expression gives us the number of moles of the gas inside the balloon.
n = (101325 Pa * 1700 m³) / (8.314 J/(mol·K) * 283.15 K)
n ≈ 8108.58 mol
Now that we know the number of moles of gas inside the balloon, we can calculate the temperature of the air inside the balloon using the same ideal gas law equation.
T = PV / nR
Substituting the known values, we have:
T = (101325 Pa) * (1700 m³) / (8108.58 mol * 8.314 J/(mol·K))
Simplifying this expression gives us the temperature (T) in Kelvin.
Finally, convert the temperature from Kelvin back to degrees Celsius to answer the question.
Using the given air pressure and outside air temperature (which is 283.15 K), we compute the air density to be:
ρ=pRT=101325287.00⋅283.15=1.24686kg/m3
This gives us:
L=mg=ρVg(ΔTT+ΔT)
780⋅9.81=1.24686⋅1700⋅9.81(ΔTT+ΔT)
(ΔTT+ΔT)=0.368
Using this ratio and T = 283.15 K, we find ΔT=165K, so T+ΔT=?