if 255mg of a radioactive substance decays to 205 mg in 49 hours, the half life of the elements is approx???

Question

expressed in hours, can someone please show me how to get to the anwser

Answer 1

With a half-life material, the basic formula for the amount of material left after time t is:
m(t) = m(0) / 2^(t / halflife)

m(0) is the initial amount of material (255 mg)
m(t) is the amount of material at time t (205 mg)
t = 49
halflife is the time when you would have half the material
In your case:

205 = 255 / 2^( 49 / h )

Now you need to solve for h:

Multiply both sides by 2^(49/h)

205 * 2^(49/h) = 255

Divide both sides by 205:
2^(49/h) = 255 / 205

Take the log base 2 of both sides:
49/h = log_2 (255/205)

Use the change of base rule log_2 (x) = log(x) / log(2)

49/h = log(255/205) / log(2)

Invert both sides:
h/49 = log(2) / log(255/205)

Multiply both sides by 49:
h = 49 * log(2) / log(255/205)

Plug that into your calculator and you should get:
half-life ≈ 155.6 hours

Answer 2

Decay of radioactive substances is described by an exponential equation:

A = Ao*exp(-k*t)

A is how much you have now, Ao is how much you started with, and t is how much time has elapsed. k is a constant that describes how fast it decays. You need to get k first.

205 = 255 * exp(-49k)

205/255 = exp(-49k)

0.804 = exp(-49k)

take ln of both sides

ln(0.804) = -49k

k = 0.00445

Now half-life means you have 50% left of the original so 255/2 = 127.5 mg left at half-life time.

Use the same equation, this time using your k value and leaving t, which will be the half-life.

A = Ao*exp(-kt)

127.5 = 255*exp(-0.00445*t)

t = 156 h