Answer:
There were 1.155 g of the substance eight hours later and it's halflife time is 1.66943 h.
Step-by-step explanation:
Since radiactive decay is expressed by a exponential function of the following kind:
mass(t) = mass(0)*e^(-a*t)
Where mass(t) is the mass of the substance at a given time, mass(0) is the inital mass substance, a is a factor that defines how fast the substance will decay and t is the elapsed time. The problem provided us with all the needed information to create the expression, with the exception of the "a". So we can apply it to the known point in order to determine "a". We have:
mass(0.5) = 32*e^(-a*0.5)
26 = 32*e^(-a*0.5)
e^(-0.5*a) = 26 / 32
e^(-0.5*a) = 0.8125
ln[e^(-0.5*a)] = ln(0.8125)
-0.5*a*ln(e) = ln(0.8125)
-0.5*a = ln(0.8125)
a = -ln(0.8125)/0.5 = 0.4152
So the expression that represents the mass of the substance over time is given by:
mass(t) = 32*e^(-0.4152*t)
After 8 hours we have:
mass(8) = 32*e^(-0.4152*8) = 32*e^(-3.3216) = 1.155 g
The half life is the time it'll take to reach half it's original mass, therefore:
16 = 32*e^(-0.4152*t)
32*e^(-0.4152*t) = 16
e^(-0.4152*t) = 16 / 32
e^(-0.4152*t) = 0.5
ln[e^(-0.4152*t)] = ln(0.5)
-0.4152*t = ln(0.5)
t = -ln(0.5)/0.4152 = 1.66943 h
