Estimate how many molecules of air are in each 2.5 L breath you take that were also in the last breath Einstein took (assumed to be the same volume). (Hint: Assume the atmosphere is about 11 km high and of constant density. Take the temperature of the air to be 27°C.)
To estimate the number of molecules of air in each breath, you need to make a few assumptions and use some basic concepts from chemistry and physics. Here's how you can approach this problem step by step:
Step 1: Calculate the total number of molecules in a 2.5 L breath.
To do this, you need to use the ideal gas law, which states that 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 in Kelvin.
First, convert the temperature from Celsius to Kelvin.
27°C + 273.15 = 300.15 K
Next, convert the volume to cubic meters.
2.5 L = 0.0025 m^3
Since the atmosphere is assumed to have a constant density, we will use the average density of the atmosphere, which is about 1.225 kg/m^3.
Calculate the mass of air in the breath using the density and volume:
mass = density * volume
mass = 1.225 kg/m^3 * 0.0025 m^3
Step 2: Convert the mass of air into moles.
To convert the mass of air into moles, you need to know the molar mass of air. Air is mostly composed of nitrogen and oxygen, so you can use the average molar mass of air, which is approximately 28.97 g/mol.
Convert the mass from kilograms to grams:
mass = 1.225 kg * 1000 g/kg
Convert grams of air to moles:
moles = mass / molar mass
Step 3: Calculate the total number of molecules in each breath.
To calculate the total number of molecules, you need to use Avogadro's number, which is approximately 6.022 x 10^23 molecules/mol.
total number of molecules = moles * Avogadro's number
Now that you have calculated the number of molecules in each breath, you need to estimate how many of these molecules were also in the last breath Einstein took.
Step 4: Estimate the number of molecules shared between each breath.
To estimate the number of shared molecules, you need to consider the height of the atmosphere and assume a constant density.
Given that the atmosphere is about 11 km high, you need to calculate the volume of air in that height.
Calculate the volume of the atmosphere:
volume = area of the base * height
Assuming the Earth's surface is approximately spherical, you can use the formula for the surface area of a sphere:
area of the base = 4πR^2
The Earth's radius is approximately 6,371 km, which is 6,371,000 meters.
Calculate the total volume of the atmosphere:
volume = 4π * (6,371,000 m)^2 * 11,000 m
Step 5: Estimate the number of molecules shared between each breath.
To estimate the number of molecules shared between each breath, you can divide the estimated volume of the atmosphere by the volume of each breath and multiply by the number of molecules in each breath.
shared molecules = (volume of the atmosphere / volume of each breath) * molecules in each breath
Finally, you will have an estimate of how many molecules of air in each 2.5 L breath you take that were also in the last breath Einstein took, assuming the same volume.
Remember that this is a rough estimate based on various assumptions, and the actual number can vary.