# Calculate the fraction of the volume that is occupied by atoms. (Hint: The atoms are spheres.)?

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The radius of a xenon atom is 1.3*10^-8cm . A 100-mL flask is filled with Xe at a pressure of 1.4 atm and a temperature of 265 K .

This is a chemistry problem/Mathematics

• First find the amount of atoms.

PV = nRT

Where

P = Pressure in Kilopascals

V = Volume in Litres

R = The general gas constant, approx 8.31 J mol^-1 K^-1

T = Temeperature in Kelvin

We want to solve for n, or mol

n = PV/RT

Now to get our units straight.

Remember that 1 atm = 101.3 kPa

100 ml = 0.1 L

1.4 atm = 141.82 kPa

265K = 265K

n = PV/RT

n = (141.82 * 0.1) / (8.31 * 265)

n = 0.00644 mol

This is the amount of xenon atoms.

The number of particles in 1 mol is equivalent to Avogadros constant, or 6.02 * 10^23 particles.

The equation is:

n = N/A

n = amount in mol

N = Number of particles in sample

n = N/A

N = nA

N = 0.00644 * 6.02 * 10^23

N = 3.88 * 10^21

Now to solve for the volume for one particle.

The volume of a sphere is given by:

V = 4/3 pi r^3

V(Xe atom) = 4/3 pi (1.3 * 10^-8)^3

V(Xe atom) = 9.2 * 10^-24 cm^3

Now to work out the volume of ALL the xenon atoms, simply multiply this by the number of particles present:

V(Xe sample) = 9.2 * 10^-24 * 3.88 * 10^21

= 0.0357 cm^3

Now to calculate the fraction of the volume these atoms occupy compared to the volume of the flask, simply divide the volume of the atoms by the volume of the flask. Take care though.

The volume of the flask, or 0.1 litres, is equivalent to 0.1 decimetres cubed. The volume of atoms calculated was in centrimetres cubed. To make the whole thing clearer, let us evaluate both volumes in METRES cubed.

In one metre cubed, there are 1*10^6 cm^3

In one decimetre cubed, there are 1*10^3 dm^3

So the volume fraction is equal to

= (0.0357 * 10^3) / (0.1 * 10^6)

= 0.000357

As a percentage this is:

0.000357 * 100 = 0.0357 % of the flask’s volume is occupied by atoms.

• Yes

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