Christopher asked in Science & MathematicsPhysics · 11 months ago

# Uncertainty on half life question?

Update:

Naturally occurring samarium includes 15.1% of the radioactive isotope 147Sm,

which decays by α-emission. One gram of natural Sm gives 89 ± 5 α decays per

second. Calculate the half-life of the isotope 147Sm, and give its uncertainty.

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• 11 months ago

Work through each step carefully and you’ll see how it’s done.

1g of natural Sm contains 15.1% of 147Sm = 1 x 15.1/100 = 0.151g of 147Sm.

Molar mass of 147Sm = 147g. Divide by Avogadro’s number to get mass of 1 atom:

m = 147/(6.022x10^23) = 2.441x10^-22g

(Alternatively, mass of a 147Sm atom is 147u = 147 x 1.6605x10^-27 = 2.441x10^-25 kg = 2.441x10^-22g.)

0.151g of 147Sm contains N atoms of 147Sm where:

N = 0.151/(2.441x10^-22) = 6.186x10^20 atoms

Activity, A = λN where λ is the decay constant. Since λ = ln(2)/T_half

A = Nln(2)/T_half

T_half = Nln(2)/A (equation 1)

Using A = 89 dps gives:

T_half = 6.186x10^20 x ln(2)/ 89 = 4.82*10^18s

You can deal with the uncertainty in 2 ways which give slightly different results.

Method 1.

A = 89±5 dps which is a fractional uncertainty of 5/89 = 0.056 = 5.6%

Since we have simply divided by A, the fractional uncertainty of the result is also 5.6%.

5.6% of 4.82*10^18s = 2.70x10^17 = 0.27*10^18s

T_half = (4.82±0.27)x10^18s

It makes sense to round the uncertainty to 1 sig. figure:

T_half = (4.8±0.3)x10^18s

(The answers are large but 147Sm has a very long half-life.)

I expect that’s the method you are meant use, but for information...

Method 2

Use equation 1 with A = 89-5 = 84dps to get an upper limit for T_half.

T_half_max = 6.186x10^20 x ln(2)/ 84 = 5.1*10^18s

Use equation 1 with A = 89+5 = 94dps to get a lower limit for T_half.

T_half_min= 6.186x10^20 x ln(2)/ 94 = 4.6*10^18s

Check my working/arithmetic of course.

• Bill
Lv 7
11 months ago

Using Avogadro's number and the given percentage 15.1 percent, calculate how many atoms of the radioactive isotope are in a one gram sample. If 89 atoms decay per second, how long would it take half of the atoms to decay?