Question

Which of the following relationship(s) is/are correct for a $1^{st}$ order reaction?

• A. $t_{75\%}=2t_{1/2}$
• B. $t_{87.5\%}=3t_{1/2}$
• C. $t_{99.9\%}=9t_{1/2}$
• D. $t_{96.87\%}=4t_{1/2}$

Answer 1

Answer: (A), (B)

$t_{87.5\%}=3t_{1/2}$

For the first order reaction:
$t=\frac{1}{k}ln\frac{A_0}{A_t}$
Half life:
$\left[A_t\right]=\left[A_0\right]/2$
$t_{\frac{1}{2}}=\frac{ln2}{k}$

For $t_{75\%}$
$\left[A_t\right]=\left[A_0\right]/4$
$t=\frac{1}{k}ln\frac{A_0}{A_0/4}$
$t=2\times \frac{1}{k}ln2$
$t_{0.75}=2\times t_{\frac{1}{2}}$

For $t_{87.5\%}$
$t_{75\%}+t_{\frac{1}{2}}=3\times t_{\frac{1}{2}}$

For $t_{99.9\%}$
For $t_{87.5\%}+t_{\frac{1}{2}}=4\times t_{\frac{1}{2}}$