How long does it take for 87.5% of each isotope to decay?

Question

Kr-73 (Half-life = 27s), Kr-74 (Half-life = 11.5 min), Kr-76 (Half-life = 14.8 hr), Kr-81 (Half-life = 2.1 x 10^5 yr)

Pleas include all steps

Answer 1

The equation is

N=N0*e^(-kt)

where N is the amount of nuclei left at time t, N0 the initial amount of nuclei and k the decay constant.

when t=half-life (let's call it t1/2) N=N0/2
Therefore
N0/2=N0*e^(-kt1/2)
Thus 2=e^(kt1/2) and k=(ln2)/t1/2

So you can rewrite the equation as

N=N0*e^(-((ln2)/t1/2)t)

Thus N0/N=e^(((ln2)/t1/2)t)
thus ln(N0/N)=((ln2)/t1/2) t

thus t= (t1/2 /ln2)*ln(N0/N)

You want to know t for the decay of 87.5%. Thus you will have 12.5% left and N=0.125N0 so you have

t=(t1/2 /ln2)*ln(N0/0.125N0)

t=(t1/2 /ln2)*ln(1/0.125)

Substitute the t1/2 for each isotope and you get the number you want in the respective order of magnitude.

This might seem more complicated than the other answers but it is more general since you can find t for any N and not just for integers of the half-life

Answer 2

in 1 half life period, 50% of the isotope decays. in the next half life, 50% of that 50% decays. therefore 50+(50/2) % of the isotope decays in 2 half lives. that is 75%. similarly, in the 3rd half life, its 50% of 25, which is 12.5%. therefore you have 50+25+12.5 % of the isotope decayed in 3 half lives. Which is 87.5%.
So the answers are:
Kr-73: 27 x 3 = 81 sec
Kr-74: 11.5 x 3 = 34.5 min
Kr-76: 14.8 x 3 = 44.4 hr
Kr-81: 2.1 x 10^5 x = 6.3 x 10^5 yrs.

Answer 3

If 87.5% of the isotope has decayed, that means that there's 12.5%, or 1/8 of the original sample left.

When a sample goes through a half life, half of the sample decays. So in 1 half life, you go from 100% of the sample to 50% of the sample, or from 1 to 1/2 (if you think in fractions).

Each subsequent half lives each decrease the population by half again. So the second half life takes you from 1/2 the original sample to 1/4 the original sample.

And the third half life takes you from 1/4 to 1/8 the original sample, which is 12.5%.

So if you must undergo 3 half lives to get to 12.5% left, then take the half life times given to you and multiply each by 3.

Answer 4

In each case three half-lives are needed

After one half-life 50% has decayed and 50% remains
After two half-lifes 75% has decayed and 25% remains
After one half-life 87.5% has decayed and 12.5% remains

This is so, whether the half life is nanoseconds or is millions of years. And whether the element is argon, krypton, neon, radon, xenon, zinc, or rhodium,
Or chlorine, carbon, cobalt, copper, tungsten, tin, or sodium

(as Tom Lehrer once sung)

So Kr-73: 81 seconds
Kr-74: 34.5 minutes
Kr-76: 44.4 hours
Kr-81: 6.3 x 10^5 years

Answer 5

just multiply each halflife by 3, because it will have first decayed a total 50% (1/2), then 75% (1/4) then 87.5% (1/8). so 81 seconds, 34.5min, etc.