The decomposition of a certain element was observed over a period of 30 hours and the results were recorded in the table below. What is the order for the reaction and what is the half life for the decomposition?
Time (hrs), % Reactant Present
0, 100%
5, 75%
10, 50%
15, 37.5%
20, 25%
25, 18.8%
30, 12.5%
it is 50% in 10 hours
so
half life is 10 hours
x = Xi e^-kt
x/Xi = 0.50 for half life
e^-kt = 0.50 when t = 10
-kt = ln 0.50 = -0.693
so
k = 0.0693
and
x/Xi = e^-0.0693 t
check at 30 hours
x/Xi = e^-0.0693*30 = e^-2.08 = 0.125 or 12.5 % wow, on the nose :)
expect so because simple result that follows decomposition in one step.
quick Google yields for example:
"............The half-life of a first-order reaction is a constant that is related to the rate constant for the reaction: t 1/2 = 0.693/ k. Radioactive decay reactions are first-order reactions. ............"
That is exactly what we have.
The 30 hour result would not have been consistent with the ten hour half life result if it were not first order.
3 half lives = 1/8 right on the nose