The general equation for exponential decay is:
#Q(t) = Q(0)e^(lambdat)#
Where Q(t) is the quantity at a given elapsed time, Q(0) is the initial quantity, and #lambda# is a decay coefficient
To find #lambda# given the half-life, you set #Q(t) = 1/2Q(0)#, t = the given time and then solve for #lambda#
#1/2Q(0) = Q(0)e^(lambda(5.27" yrs"))#
Divide both sides by Q(0):
#1/2 = e^(lambda(5.27" yrs"))#
To make the exponential function disappear, we use the natural logarithm:
#ln(1/2) = lambda(5.27" yrs")#
Replace #ln(1/2)# with -ln(2)
#-ln(2) = lambda(5.27" yrs")#
#lambda = -ln(2)/(5.27" yrs")#
Now we can evaluate at the equation at #t= 60.50" yrs"# and #Q(0) = 1" g"#
#Q(60.50" yrs") = (1" g")e^(-ln(2)/(5.27" yrs")60.50" yrs")#
#Q(60.50" yrs") = (1" g")e^(-ln(2)/(5.27" yrs")60.50" yrs")#
#Q(60.50" yrs") = 3.50xx10^-4" g"#
