Answer to Question #146834 in Differential Equations for Nikhil

Question

Uranium disintegrates at a rate3 proportional to the amount present at any instant. If m1 and m2 grams of uranium are present at time t1 and t2 , respectively, show that half life of uranium is (t1-t2)In2/In(m1/m2)

Answer 1

$\nu=\frac mM=\frac{N}{N_A},$ $m \sim M.$

$\frac{m}{m_0}=2^{-\frac tT},$

$log_2=-\frac tT,$

$T=-\frac{t}{log_2\frac{m}{m_0}}=\frac{t}{log_2\frac{m_0}{m}}=\frac{tln2}{ln\frac{m_0}{m}},$

$\frac{lnm_0-lnm}{tln2}=\frac 1T,$

$lnm_0=\frac{tln2}{T}+lnm=const,$

$\frac{t_1ln2}{T}+lnm_1=\frac{t_2ln2}{T}+lnm_2,$

$\frac{(t_1-t_2)ln2}{T}=lnm_2-lnm_1,$

$T=\frac{(t_1-t_2)ln2}{ln\frac{m_2}{m_1}}.$