Explanation:
The given data is:
The half-life of gentamicin is 1.5 hrs.
The reaction follows first-order kinetics.
The initial concentration of the reactants is 8.4 x 10-5 M.
The concentration of reactant after 8 hrs can be calculated as shown below:
The formula of the half-life of the first-order reaction is:
[tex]k=\frac{0.693}{t_1_/_2}[/tex]
Where k = rate constant
t1/2=half-life
So, the rate constant k value is:
[tex]k=\frac{0.693}{1.5 hrs}[/tex]
The expression for the rate constant is :
[tex]k=\frac{2.303}{t} log \frac{initial concentration}{concentration after time "t"}[/tex]
Substitute the given values and the k value in this formula to get the concentration of the reactant after time 8 hrs is shown below:
[tex]\frac{0.693}{1.5 hrs} =\frac{2.303}{8 hrs} x log \frac{8.4x10^-^5}{y} \\ log \frac{8.4x10^-^5}{y} =1.604\\\frac{8.4x10^-^5}{y}=10^1^.^6^0^4\\\frac{8.4x10^-^5}{y}=40.18\\y=\frac{8.4x10^-^5}{40.18} \\=>y=2.09x10^-^6[/tex]
Answer: The concentration of reactant remains after 8 hours is 2.09x10^-6M.
