half life question i'm struggling with

Question

5. Technetium-99 is a gamma emitter with a half-life of 6 hours. It is used for imaging of heart function. A patient receives an injection containing 12 mg of Technetium-99 at 6:00 a.m. on Tuesday. How much remains at 9:00 p.m. the next day.

Answer 1

The previous answer is fine, but I thought I'd provide another approach that you might find easier to work with.

Fraction remaining = 0.5ⁿ where n = number of half lives that have elapsed. That's the formula I use.

The time elapsed is 6am - 6am = 24 hours + 6 am - 9 pm = 15 hours = 39 hours

Number of half lives elapsed = 39 hours x 6 hours/half life = 6.5 half lives elapsed. Plug this into the formula..

Fraction remaining = 0.5ⁿ = 0.5^6.5 = 0.01105

To find the mass remaining: 12 mg x 0.01105 = 0.13 mg remaining

Answer 2

I used the integrated rate law for this:

ln([Tech intial]/[Tech final]) = -kt

where k is a constant and t is time in hours. A half life is the time it takes for half of the stuff to react. Thus,

the above equation becomes:

ln(2) = -kt or

t = 0.693/k

since the half life is 6 hours,

6 = 0.693/k

k = 0.1155

ln([Tech intial]/[Tech final]) = -kt

now you know the value of k and your initial Tech amount, 12 mg. Plug them in:

ln(12/[Tech final]) = -0.115*t

plug in your time and solve for Tech final.

Unsure if what you meant by "the next day."