Question

Carbon dioxide dissolved in soda can escape from the PET bottle slowly by means of diffusion. How long does it take for the soda to lose 10% of its original concentration? The following information might be useful: The inside surface of the bottle can be assumed to be saturated with carbon dioxide. The outside surface concentration can be neglected. It can be assumed that diffusion only takes place through the walls of the cylinder and not through the top or bottom. The bottle can be considered as a cylinder.

Diffusivity of CO2 in PET = 2×10-13 m2 /s;

Inner diameter of the bottle = 0.08 m;

Height of the bottle = 0.2 m;

Solubility limit of CO2 in PET = 10 kg CO2 per m3 of PET;

PET thickness = 0.5 mm;

Initial concentration of CO2 in cola = 7 g/L

Answer 1

SOLUTION:

Data given in the question:

Diffusivity of CO2 in PET = 2×10-13 m2 /s

Inner diameter of the bottle = 0.08 m

Height of the bottle = 0.2 m

Solubility limit of CO2 in PET = 10 kg CO2 per m3 of PET

PET thickness = 0.5 mm

Initial concentration of CO2 in soda = 7 g/L

Assuming that the PET bottle is filled to its capacity, volume of the PET bottle = volume of soda

Volume of a cylinder = π*r2*h

Inner diameter of the bottle = 0.08 m

Inner radius of the bottle = 0.04 m

Height of the bottle = 0.2 m

Volume of soda in the bottle = π*0.042*0.2 = 1.005*10-3 m3 = 1.005 L

Therefore, initial concentration of CO2 in soda bottle = 7*1.005 = 7.035 g

According to Winkelmann's equation, the time taken for diffusion,

t = {ρ*yBM*(Zt2-Z02)}/{2*M*C*DAB*(yA1-yA2)}

where,

ρ = Density of soda water = Concentration of CO2*Density of CO2 + Concentration of water*Density of water

= 0.007*.00198 g/cm3 + 0.993*1 g/cm3 = 0.993 g/cm3

yA1 = Concentration of CO2 inside the bottle = 2.859*10-3 (mole frac)

yA2 = Concentration of CO2 outside the bottle = 10% of original concentration = 2.859*10-4 (mole frac)

yBM = ln((1-yA2)/(1-yA1)) = ln((1-2.859*10-4)/(1-2.859*10-3)) = 2.577*10-3

DAB = Diffusivity of CO2 = 2×10-13 m2 /s

C = P/R*T

Assuming standard atmospheric conditions, C= 1/0.0821*298 =0.04 mol/L = 40 mol/m3

M = Molecular weight of soda water = 62

Zt2-Z02 = PET thickness through which diffusion occurs

PET thickness = 0.5 mm

Therefore, Zt2-Z02 = (0.5*10-3)2 = 2.5*10-7 m2

Substituting all these values, we get

t = {ρ*yBM*(Zt2-Z02)}/{2*M*C*DAB*(yA1-yA2)} = 27048 sec = 7.51hours (approx)