Thinking like a molecule: As mentioned in class on several occasions the air we
ID: 946897 • Letter: T
Question
Thinking like a molecule: As mentioned in class on several occasions the air we breathe everyday can be treated as ideal gas. Calculate the average distance between gas molecules in air at room temperature (25 C) and 1 atmosphere pressure. Calculate the average distance between water molecules (liquid state) at room temperature. Take the density of water to be 1 g/cc. If we inhale 1 liter of air with every breath, how many molecules of oxygen do we inhale with every breath? (assume air is 19% oxygen) How fast are nitrogen molecules moving on average?Explanation / Answer
a.
There are 6.02x 1023 molecules of gas in one mol which occupies 22.4 liters at atmospheric pressure and temperature of 0C.
This is a density; call it n, of 6 x 1023/22.4, or 2.7 x 1022 molecules per liter. A liter is 1000 cm3, for example a cube 10 cm (0.1m) on a side.
If the molecules were equally spaced, then n1/3, or 3.0 x 107 is the average number of molecules along one edge of the cube.
The spacing in cm. is then 10/n1/3, or 3.3 x 10-7 cm, or 3.3 x 10-9 m. Compare this with the size of the molecules. A good reference size for a small atom like hydrogen is 1 Angstrom (this is written as 1 Å) which is 10-10 m.
Oxygen molecule is about 5 Å in diameter. Therefore the molecules in air are roughly 8 times as far apart as the the width of the individual molecules.
b.
1 g/cm^3 * 1mole/18 g (note grams cancel) * 6.02 E23 molecules/mole (moles cancle)
= .32 E23 molecules/ cm^3 = .032 E24 molecules/cm^3
cube root above
.32 E8 /cm
oops invert above
cm/.32 E8 = 3.3 atoms per 10^-8 cm
3 atoms/ angstrom is estimate
c.
1 liter of air at 19% oxygen possesses 0.19 L of oxygen. Since at STP 1 mole of gas occupies 22.4 L, simply divide 0.19/22.4, to arrive at 0.00848 moles of oxygen.
d.
Boltzmann relation:
E = 1/2 m v² = 3/2 k T or v = (3 k T / m)
where m is the mass of nitrogen molecule: m = Mu = 28 x 1.66 x 10^-27 kg and v = (3 x 1.38 x 10^-23 J/K x 293 K / 28 x 1.66 x 10^-27 kg) = 510 m/s.
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