Question

A formulation has an initial degradant at a concentration of 0.02% (% weight). The regulatory limit for that degradant is 0.4%. The measured level of that degradant is 0.42% at 40oC after 5 months, and 0.40% at 60oC after 1 month. Determine the shelf-life of this product at 25oC .

Answer 1

Given data:

The following values are given in the question.

1. Initial degradant concentration = 0.02%
2. Regulatory limit = 0.4%
3. Degradant at 40oC after 5 months = 0.42%
4. Degradant at 60oC after 1 month = 0.40%
5. Shelf life of the product = ?

Solution:

Step -1: Calculate rate of change of impurity (K)

K can be calculated using the following equation;

K = (Level of impurity of defined time point - Initial level of impurity) / Time point (Number of months) ----> 1

a. Rate of change of impurity at 60oC:

Substitute the values in equation 1;

K1 = (0.40 - 0.02) / 1 = 0.38 / 1 = 0.38% per month.

b. Rate of change of impurity at 40oC:

Substitute the values in equation 1

K2 = (0.42 - 0.02) / 5 = 0.40 / 5 = 0.08% per month.

Hence the rate of degradation at 60oC, K1 = 0.38% and at 40oC, K2 = 0.08%

Step-2: Calculate activation energy (Ea)

Calculate Ea using the following equation derived from Arrhenius equation of both the storage conditions.

Ea = R x T1 x T2 [ln K1 - ln K2] / [T1 - T2] ----------> 2

where R is universal gas constant = 1.987cal / K.mol; T1 is temperature in K for K1 =  333K; T2 is temperature in K for K2 =  313K; K1 is rate of change of impurity at 60oC = 0.38%; K2 is rate of change of impurity at 40oC = 0.08%

Substitute the values in equation 2 and solve

Ea = (1.987cal / K.mol)(333K)(313K)[ln 0.38 - ln 0.08] / [333K - 313K]

= 207103 cal K / mol [-09676 - (-2.5257)] / 20K

= (207103 cal / mol x 2.4289) / 20

= 503032.48 cal / mol / 20 = 25151.62 cal / mol

Hence the Ea required for the degradation is 25151.62 cal / mol

Step - 3: Calculate the rate Constant (A)

Calculate the A using the following equation derived from Arrhenius equation of both the storage conditions.

ln A = ln K1 + T2 [ln K1 - ln K2] / [T1 - T2] -------> 3

where A is rate constant; T1 is temperature in K for K1 =  333K; T2 is temperature in K for K2 =  313K; K1 is rate of change of impurity at 60oC = 0.38%; K2 is rate of change of impurity at 40oC = 0.08%

Substitute the values in equation 3 and solve

ln A = ln 0.38 + 313K [ln 0.38 - ln 0.08] / [333K - 313K]

=  -09676 + 313 [-09676 - ((-2.5257)] / 20

= (312.9 x 2.4289) / 20 = 760.01 / 20 = 38.0

Hence ln A is 38.0

Step - 4: Calculate the rate of change in impurity (K3) at 25oC

Calculate the K3 using the following equation derived from Arrhenius equation.

ln K3 = ln A - [Ea / RT3] ------> 4

where K3 is rate of change in impurity at 25oC; R is universal gas constant = 1.987cal / K.mol; T3 is temperature in K for K3 =  298K; ln A is rate constant = 38.0; Ea is activation energy = 25151.62 cal / mol

Substitute the values in equation 4 and solve

ln K3 = 38.0 - [(25151.62 cal / mol) / (1.987cal / K.mol x 298K)]

= 38.0 - [(25151.62 cal / mol) / (592.13 cal / mol]

= 38.0 - 42.48 = - 4.48

K3 = 0.0113% per month

Hence at 25oC the rate of change of impurity is 0.0113% per month.

Step - 5: Calculate the shelf life

Shelf life = [Specification limit of impurity - Initial level of impurity] / Rate of the change in impurity at 25oC / month

Substitute the values in the above equation and solve

Shelf life = [0.40% - 0.02%] / 0.0113% per month = 0.38 / 0.0113 month-1 = 33.6 month = 34 months.

Hence the shelf life of the product at 25oC is 34 months.