How much time will it take for 30 grams of 222Rn (222 as in the atomic Mass of Rn) to decay to 7.5 g?

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• Dr W
Lv 7
1 month ago

you need to know the half life of 222Rn to solve this

.. half life = 3.8235(3) days

fyi.. that (3) means the 3.8235 is a measured number and 1 standard deviation of the measurements is 3 in the rightmost digit... .i.e.. 67% of the time, the measurement falls between 3.8232 and 3.8238 days.

anyway, you can either

.. (1) eyeball this and see that 30g/2 = 15g (1 half life).. 15g/2 = 7.5g (2nd

.. . . .half life) so that 2 half lives have passed so that

.. . .. . .t = 2*3.8235 days = 7.647 days

OR...

.. (2) the more general approach.. . showing all the steps 1 at at time

.. . .. ..... A(t) = A(o) * (1/2)^(t / half life).. .. .<----- memorize this

.. . .. ..... A(t) / A(o) = (1/2)^(t / half life)

.. . .. ..... ln(A(t) / A(o)) = (t / half life) * ln(1/2)

.. . .. ..... ln(A(t) / A(o)) = (t / half life) *(ln(1) - ln(2))

.. . .. ..... ln(A(t) / A(o)) = (t / half life) *(0 - ln(2))

.. . .. ..... ln(A(t) / A(o)) = (t / half life) * (-ln(2))

.. . .. ..... ln(A(t) / A(o)) / -ln(2) = (t / half life)

.. . .. ..... ln(A(o) / A(t)) / ln(2) = (t / half life)

. . .. . and finally

.. .. ... .. .t = half life * (ln(A(o) / A(t)) / ln(2)

.. .. ... .. .t = half life * (ln(30 / 7.5) / ln(2)

.. .. ... .. .t = half life * ln(4) / ln(2)

.. .. ... .. .t = half life * ln(2² ) / ln(2)

.. .. ... .. .t = half life * 2*ln(2 ) / ln(2)

.. .. ... .. .t = half life * 2

.. .. ... .. .t = 3.8235 days * 2

.. .. ... .. .t = 7.647 days

• 1 month ago

2 half lives:

30 * 1/2 = 15 * 1/2 = 7.5

• 1 month ago

(7.5 g) / (30 g) = 0.25 = 1/4 = (1/2)^2

So takes two half-lives, but that's not really a good measure of time.

So looking up the half-life of 222Rn in the source below:

3.82 days x 2 = 7.64 days

• 1 month ago

²²²Rn has a half life of 3.82 d

Half life

N(t) = N₀(1/2)^(t/th)

N₀ is initial amount

N(t) is the amount remaining after time t

th is the half life time, ie, time for half the amount to decay

7.5 = 20(1/2)^(t/3.82)

(1/2)^(t/3.82)  = 7.5/20 = 0.375

t/3.82 = log (base0.5)0.375 = 1.415

t = 5.41 days

• Anonymous
1 month ago

7.5 g is 34% of the original amount

0.34 = 0.5^n, solve for n which is the number of half lives

time = n * half life for Radon.