The​ half-life of radium

Question

The​ half-life of radium is 1690 years. If 20 grams are present​ now, how much will be present in 50 ​years?

Answer 1

N= N₀ e ^-λt

Where λ = Decay constant

N₀ = 20 gm (Original Value )

N= ? ( left after given time, t = 50 years )

t= 50 years, λ = decay constant

Half life = 0.693/ λ

λ= 0.693/Half life

λ = 0.693 /1690= 4.1*10^-4

e^{- λt} =e ^{- 4.1*10-4 *50} = e ^{- 2.05* 10-2}

N = 20*e ^{- 2.05* 10-2} = 20/e ^{2.05* 10-2} = 20/ 1.02 = 19.61 gm

After 50 years 19.61 gm will be left

Answer 2

I like to set an expectation so I can interpret and verify my final answer later. If the half-life of radium is 1690 years, that would mean that after 1690 years, 20 g of radium would decay into (1/2)(20 g) = 10 g of radium.

50 years is much shorter than 1690 years, so I'd predict a mass less than 20g and much closer to 20g than 10g. Let's see if the math bears that out!

50 years is much shorter than a full half-life of 1690 years, so we would only halve the radium 50/1690ths of a time...if that wording makes sense! In my opinion, the mathematical expression is easier to interpret:

20 g * (1/2)^(50/1690) = 19.594 g radium