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?

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Answer 1 · 2017-07-16T13:27:45

Starting with the equation:

#Q(t) = Q(0)e^(kt)#

Substitute the value of #t = 8" min"#:

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

Divide both aides of the equation by #Q(0)#:

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

You use the fact that #(Q(t_"half-life"))/(Q(0)) = 1/2#

#1/2=e^(k8" min")#

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

#ln(1/2)=ln(e^(k8" min"))#

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

#-ln(2)=k8" min"#

Flip the equation and divide by #8" min"#

#k = -ln(2)/(8" min")#