2 One item that Detective Man noticed was that on the table next to the body was

Question

2 One item that Detective Man noticed was that on the table next to the body was an invention, supposedly invented by Mr. Tim; the invention was called "An Ever-Fresh Coffee Maker". What the invention would do was start off with a pot of coffee containing 64 ounces (8 cups) of coffee containing 1300 mg of caffeine. Coffee containing 125 mg of caffeine per 5 ounces was pumped into the 64 ounce pot at a rate of 6 ounces per minute; after the new mixture is well mixed, the well-mixed coffee is then pumped out of the pot at a rate of 4 ounces per minute. The idea was to have more caffeine and more coffee ready as the day went on. Detective Man wondered how much caffeine was in the pot now 30 minutes after he had inadvertently turned it on when he had tripped over the corpse. How much is there? 3 When the initial autopsy was done at 2:00 AM one hour after the body was found (Detective Man is an amateur medical examiner who carries his instruments with him at all times, there was found 100 grams of sodium-24, which has a half-life of15 hours. Thus Detective Man knew what the original amount of sodium-24 was in the body when Mr. Tim was killed because he knew the time of death and assumed that the sodium started decaying at the time of death). How much was there?

Explanation / Answer

3. Death occurred at 9.52 PM and the autopsy was done at 2:00 AM; time elapsed = 4 h 8 minutes = 4 h + (8 minutes)*(1 h/60 mins) = (4 hrs + 0.1333 hrs) = 4.1333 h.

Use the law of radio-active disintegration

Nt = N0*e-kt where Nt = amount of a radioactive substance present at time t and N0 = amount of the substance present initially; k is the radio-active decay constant and is given by

k = 0.693/t1/2 where t1/2 is the half life of the radio-active element.

Convert the expression to a more useable form:

Nt/N0 = e-kt

ln (Nt/N0) = -kt

Put Nt = 100 g, k = 0.693/15 h-1 (since t1/2 = 15 h) and t = 4.1333 h; therefore,

ln (100 g/N0) = -(0.693/15 h-1)*(4.1333 h) = -0.191

===> (100 g/N0) = e^(-0.191) = 0.826

===> N0 = (100 g)/(0.826) = 121.065 g

The initial amount of sodium-24 present was 121.065 g (ans).

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