# A lab has 40 kg of radioactive uranium this substance has a half-life of 8 minutes and decays exponentially. How do you find the decay constant k?

Jul 16, 2017

Starting with the equation:

$Q \left(t\right) = Q \left(0\right) {e}^{k t}$

Substitute the value of $t = 8 \text{ min}$:

$Q \left(8 \text{ min") = Q(0)e^(k8" min}\right)$

Divide both aides of the equation by $Q \left(0\right)$:

(Q(8" min"))/(Q(0))=e^(k8" min")

You use the fact that $\frac{Q \left({t}_{\text{half-life}}\right)}{Q \left(0\right)} = \frac{1}{2}$

$\frac{1}{2} = {e}^{k 8 \text{ min}}$

We can eliminate the exponential by using the natural logarithm on both sides:

$\ln \left(\frac{1}{2}\right) = \ln \left({e}^{k 8 \text{ min}}\right)$

We know that ln(1/2) = -ln(2) and the inverses will disappear on the right:

$- \ln \left(2\right) = k 8 \text{ min}$

Flip the equation and divide by $8 \text{ min}$

$k = - \ln \frac{2}{8 \text{ min}}$