Cocaine is a weak base (Figure see below) with pKb = 5.59. Cocaine has local anesthetic effect (local anesthetic) but is also a narcotic.
a) Calculate the ratio of the concentration of cocaine and its conjugated acid in a solution which has. pH = 7.40 (that of blood) ii. pH = 1.50 (that of the stomach)
b) If cocaine have the best solubility in an aqueous solution with pH 7.40 or pH = 1.50. Explain your answer.
a) To calculate the ratio of the concentration of cocaine and its conjugated acid, we can use the Henderson-Hasselbalch equation:
pH = pKa + log ([A-]/[HA])
In this case, the conjugated acid is HA (cocaine) and the conjugate base is A- (cocaine's ionized form). Since cocaine is a weak base, it will mainly exist in its unionized form (conjugated acid) at lower pH (such as in the stomach) and in its ionized form (conjugate base) at higher pH (such as in blood).
i. pH = 7.40 (blood):
Here, we can use the pKb value to calculate pKa:
pKa = 14 - pKb = 14 - 5.59 = 8.41
Now we can substitute the values into the equation:
7.40 = 8.41 + log ([A-]/[HA])
Solving for the ratio ([A-]/[HA]):
[A-]/[HA] = 10^(7.40 - 8.41) = 10^(-1.01) = 0.079
Therefore, in a solution with pH 7.40, the concentration of cocaine's conjugate base is 0.079 times the concentration of cocaine's conjugated acid.
ii. pH = 1.50 (stomach):
Using the same process, we can calculate pKa by:
pKa = 14 - pKb = 14 - 5.59 = 8.41
Substituting the values in the Henderson-Hasselbalch equation:
1.50 = 8.41 + log ([A-]/[HA])
Calculating the ratio ([A-]/[HA]):
[A-]/[HA] = 10^(1.50 - 8.41) = 10^(-6.91) = 9.18 x 10^(-7)
Hence, in a solution with pH 1.50, the concentration of cocaine's conjugate base is 9.18 x 10^(-7) times the concentration of cocaine's conjugated acid.
b) To determine in which pH cocaine has the best solubility, we need to consider its ionization state. Cocaine is more soluble when it is in its ionized form (conjugate base). Therefore, it will have better solubility in a solution with pH = 7.40 (blood) when compared to pH = 1.50 (stomach). In blood, the higher pH will favor more ionization of the cocaine molecule, making it more soluble.
In contrast, in the stomach (pH = 1.50), the low pH conditions will favor the protonation of cocaine, resulting in a higher concentration of the unionized (conjugated acid) form, which is less soluble in water.
In summary, cocaine has better solubility in an aqueous solution with pH 7.40 (blood), as it favors the ionized form (conjugate base) and higher solubility.