Question

The radius of a Xenon atom is 1.3 X10^-8 cm. A 100 mL flask is filled with Xe at a pressure of 1.0 atm and a temperate of 287 K. Calculate the fraction of the volume that is occupied by Xe atoms. (Hint. The atoms are spheres.) express answer using two significant figures.

Answer 1

Given radius of Xe atom, r=1.3x10^-8 cm

Pressure, P=1 atm, volume, V=100 m=0.1 L, temperature, T=287 K.

From ideal gas equation, PV=nRT, where R=0.0821 L atm mol^-1 K^-1.

Number of moles, n=PV/RT

n=(1 atm x 0.1 L)/(0.0821 L atm mol^-1 K^-1 x 287 K)

n=4.24x10^-3 mol.

But one mole=6.023x10^23 atoms.

Therefore number of Xe atoms=6.023x10^23 atoms x 4.24 x10^-3 =2.556x10^21 atoms.

Since the Xe atoms are spheres, therefore volume of sphere,

V=(4/3)πr^3

V=(4/3) x 3.14 x (1.3x10^-8 cm)^3

V=9.198x10^-24 cm^3=9.198x10^-24 mL

Total volume occupied for Xe atoms =V x number of Xe atoms

=9.198x10^-24 mL x 2.556 x 10^21 = 0.0235 mL.

Fraction of volume that occupied by Xe atoms=volume occupied by Xe atoms/total volume=0.0235 mL/100 mL=0.0235%.

Fraction of volume occupied by Xe atoms=0.0235%.

Please let me know if you have any doubt. Thanks.