Question

Use the Kelvin equation to calculate the radius of an open-ended tube within which capillary condensation of nitrogen at 77 K and a relative pressure of 0.75 might be expected. Assume that prior adsorption of nitrogen has formed a layer 0.9 nm thick coating the inside of the tube. For nitrogen at 77 K the surface tension is 8.85 mN m-1 and the molar volume is 34.7 cm3 mol-1 .

Answer 1

Solution:- kelvin equation is, ln(p/p0) = 2yV_m/(r.RT)

where, p = actual vapor pressure, p0 = saturated vapor pressure

y = surface tension, V_m = molar volume, r = radius

R = universal gas constant and T = kelvin temperature

p/p0 is known as relative pressure and it's value is given as 0.75.

Value of R is 8.314 J.mol^-1.K^-1

since, J = N.m

So, R = 8.314 N.m.mol^-1.K^-1

Molar volume = 34.7 cm³.mol^-1 x (1m)³/(100 cm)³ = 3.47 x 10^-5 m³.mol^-1

surface tension = 8.85 mN.m^-1 x (1 N/1000 mN) = 8.85 x 10^-3 N.m^-1

let's plug in the values in the equation...

ln(0.75) = 2 x 8.85 x 10^-3 N.m^-1 x 3.47 x 10^-5 m³.mol^-1/(r x 8.314 N.m.mol^-1.K^-1 x 77 K)

value of ln(0.75) is -0.29. We could neglect the negative sign here since the radius is not like a negative thing.

0.29 = 9.59 x 10^-10 m/r

r = 9.59 x 10^-10 m/0.29 = 3.31 x 10^-9 m

1 m = 10⁹ nm

So, the radius, r = 3.31 nm