What is the formula for the quantity of radium left after t years?

[Jacobpm64]

(a) The half-life of radium-226 is 1620 years. Write a formula for the quantity, Q, of radium left after t years, if the initial quantity is Q₀.

Check me on this one:
Q = (Q₀ / 2^(1620/t))

(b) What percentage of the original amount of radium is left after 500 years?

Check this one as well:
Q = (Q₀ / 2^1620/500)
Q = (Q₀ / 2^3.24)
Q = Q₀ * 2^-3.24
Q = Q₀(0.1058)
10.6%

 

[Astronuc]

In part 'a', the exponent on the 2 should reflect the number of half-lives, which is the ratio of t/T_1/2, where T_1/2 is the half-life.

So after 1 half-life, Q/Q_o= 1/2, and after two half-lives, Q/Q_o= (1/2)² = 1/4, . . .

In the part 'b', the half-life of Rn-226 is 1620 years, the point at which 50% would be remaining, and 500 years is less than 1/3 of the half-life, so does 10.6% look right?

 

[HallsofIvy]

In other words, you have the exponent "upside down". It should be
t/1620, not 1620/t.

 

[Jacobpm64]

how's this?

(a) Q = (Q₀ / 2^t/1620)

(b) Q = (Q₀ / 2^500/1620)
Q = (Q₀ / 2^0.3086...)
Q = Q₀ * 2^-0.3086...
Q = Q₀(0.8074...)
80.7%

 

[Astronuc]

Better.

 

[HallsofIvy]

Do you understand why it is t/1620 rather than 1620/t? The "half life" of a substance is the time it takes to degrade to half its original value. Every time one "half life" passes, the amount is multiplied by 1/2: if the original amount is M, after one "half life" the amount is (1/2)M. After a second "half life", it is (1/2)((1/2)M)= (1/2)²M. After a third "half life" we multiply by 1/2 again: (1/2)((1/2)²M)= (1/2)³M. That is, the exponent just counts the number of "half lives" in the t years. If the "half life" is 1620, that "number of half lives" is t/1620 so the amount will be (1/2)^t/1620M= M/2^t/1620.