Enrico Fermi was a famous physicist who liked to pose what are now known as Fermi problems in which several assumptions are made in order to make a seemingly impossible estimate. One example of a Fermi problem is "Caesar's last breath" which estimates that you, right now, are breathing some of the molecules exhaled by Julius Caesar just before he died.
Assumptions:
1. The gas molecules from Caesar's last breath are now evenly dispersed in the atmosphere.
2. The atmosphere is 50 km thick, has an average temperature of 15 °C, and an average pressure of 0.20 atm.
3. The radius of the Earth is about 6400 km.
4. The volume of a single human breath is roughly 500 mL.
Perform the following calculations, reporting all answers to two significant figures.
Calculate the total volume of the atmosphere.
Calculate the total number of gas molecules in the atmosphere.
Calculate the number of gas molecules in Caesar's last breath (37°C and 1.0 atm).
What fraction of all air molecules came from Caesar's last breath?
About how many molecules from Caesar's last breath do you inhale each time you breathe?
Your question is not correct . I cannot understand your question
It is not one question. It is a series of questions. First calculate the total volume of the atmosphere. Second calculate the total number of gas molecules in the atmosphere. Third calculate the number of gas molecules in Caesar's last breat. Fourth calculate what fraction of all air molecules came from Caesar's last breath. Lastly, calculate how many molecules from Caesar's last breath do you inhale each time you breathe.
Perhaps I can get you started.
Wouldn't the volume of the atmosphere+earth = (4/3)*pi*r^3. r would be radius of earth + thickness of atmosphere for total volume. Then determine volume of earth and subtract to obtain volume of the atmosphere.
I keep getting the answer to the volume of the atmosphere wrong. Are my calculations correct? Or what is my problem?
I calculate that the total volume of Earth + thickness of atmosphere is:
(4/3)*pi*(6450000 m)^3 = 1.124*10^21
I then calculate that the volume of the Earth is:
(4/3)*pi*(6400000 m)^3 = 1.098*10^21
Finally I subtract the volume of Earth from the total volume, which gives me that the volume of the atmosphere is 2.6*10^19
you only answered one part of the question!
Oh, Caesar's last breath, huh? That's some heavy breathing!
Let's tackle these calculations and see what comes out:
First, let's calculate the total volume of the atmosphere:
The atmosphere is 50 km thick, so its volume is the area of Earth's surface multiplied by the thickness:
Volume = π * (radius of Earth)^2 * thickness of atmosphere
Volume ≈ 3.14 * (6400 km)^2 * 50 km
Calculating... drumroll, please... and we get a total volume of the atmosphere around 5.10 x 10^14 cubic kilometers. That's a lot of air for such a tiny planet!
Now, let's calculate the total number of gas molecules in the atmosphere:
We'll use the ideal gas law here. The pressure, volume, temperature, and number of gas molecules are related through this formula:
PV = nRT (P is pressure, V is volume, T is temperature, n is the number of gas molecules, and R is the ideal gas constant)
We're given the average pressure and temperature, so let's plug those in:
n = (PV) / (RT)
n ≈ (0.20 atm) * (5.10 x 10^14 cubic kilometers) / (0.0821 L * atm / K * mol) * (288 K)
Calculating... and we have around 2.5 x 10^44 gas molecules in the atmosphere. Talk about being overcrowded!
Next, let's calculate the number of gas molecules in Caesar's last breath (at 37°C and 1.0 atm):
We can use the same formula as before, but this time we know the pressure and temperature of Caesar's breath, and the volume of a single breath. We just need to solve for n:
n = (PV) / (RT)
n ≈ (1.0 atm) * (500 mL) / (0.0821 L * atm / K * mol) * (310 K)
Calculating... and we find that there are approximately 3.0 x 10^21 molecules in Caesar's last breath. A good reminder that no one breathes alone, even in the afterlife!
Now, let's determine what fraction of all air molecules came from Caesar's last breath:
Fraction = (Number of molecules in Caesar's breath) / (Number of molecules in the atmosphere)
Fraction ≈ (3.0 x 10^21 molecules) / (2.5 x 10^44 molecules)
Calculating... and we see that it's a teeny, tiny fraction. The fraction of air molecules from Caesar's last breath is about 1.2 x 10^-23, which is practically zero when compared to the total number of molecules in the atmosphere. Sorry, Caesar, your breath doesn't have much staying power!
Finally, let's estimate how many molecules from Caesar's last breath you inhale with each breath:
We need to know how many molecules are in the air you inhale per breath. Assuming your lung capacity is about 500 mL, we can use the ideal gas law once more:
n = (PV) / (RT)
n ≈ (0.20 atm) * (500 mL) / (0.0821 L * atm / K * mol) * (288 K)
Calculating... and we find that you inhale approximately 2.4 x 10^19 molecules from Caesar's last breath with each breath. It's like getting a little historical boost every time you inhale!
I hope these calculations cleared the air for you. Don't forget to exhale the laughter!
To solve this Fermi problem, we'll take the following steps:
Step 1: Calculate the total volume of the atmosphere.
Since the atmosphere's radius is not given, we can approximate it by assuming a constant thickness of 50 km. Therefore, the volume of the atmosphere can be calculated as the area of a circle with a radius of 6400 km plus a 50 km thickness.
Volume = (4/3) * π * (radius + thickness)³
= (4/3) * π * (6450 km)³
Now we need to convert this volume from km³ to mL since the volume of a breath is given in mL. To convert, we multiply by 10^15 to account for the conversion factor from km³ to mL.
Total volume of the atmosphere = (4/3) * π * (6450 km)³ * (10^15 mL/km³)
Step 2: Calculate the total number of gas molecules in the atmosphere.
To calculate the number of gas molecules, we need to multiply the volume of the atmosphere by the number of molecules per unit volume. Assuming the ideal gas law, we have the formula:
P * V = n * R * T,
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.
Using this equation, we can rearrange it to solve for n (number of moles):
n = P * V / (R * T)
Now we multiply the number of moles by Avogadro's constant to get the number of gas molecules:
Number of gas molecules in the atmosphere = (P * V / (R * T)) * Avogadro's constant
Step 3: Calculate the number of gas molecules in Caesar's last breath.
To calculate this, we will use the same ideal gas law formula used in step 2. However, we will use the given conditions for the temperature and pressure of Caesar's last breath.
Step 4: Calculate the fraction of all air molecules that came from Caesar's last breath.
To calculate this, we divide the number of gas molecules in Caesar's last breath by the total number of gas molecules in the atmosphere.
Fraction of air molecules from Caesar's last breath = (Number of gas molecules in Caesar's breath) / (Number of gas molecules in the atmosphere)
Step 5: Calculate the number of molecules from Caesar's last breath in each breath you take.
To calculate this, we divide the number of gas molecules in Caesar's breath by the average volume of a single breath.
Number of molecules from Caesar's last breath in each breath = (Number of gas molecules in Caesar's breath) / (Volume of a single breath)
Using these steps, you can now perform the calculations and report the answers to two significant figures.