How do I calculate the time taken for an element to decay from one mass to another?

[kkid]

TA sample of ²⁴Na has a half-life of 234 hours, How much time elapses before a 5μg sample contains 1μg of undecayed atoms?



I have calculate the decay constant (8.23x10^-7)


What do I do now?

 

[SammyS]

TA sample of ²⁴Na has a half-life of 234 hours, How much time elapses before a 5μg sample contains 1μg of undecayed atoms?

I have calculate the decay constant (8.23x10^-7)

What do I do now?

Hello kkid. Welcome to PF !

What's the formula for radioactive decay?

 

[kkid]

I eventually did this by altering this equation:

N = N₀e^-λt

if one divides by N₀ then the left hand side is just a faction of the start and end amounts (whether this is in terms of amount of atoms or in terms of mass it will be the same.

In this case the left hand side of the equation (the fraction) will be 1/5 as 1 is 1/5 of 5.


This gives me 1/5 = e^-λt

taking ln of both sides to eliminate e gives:

ln(1/5) = -λt

t = ln(1/5)/-λ


I substitute the value of decay constant (λ) to get my overall answer of 543.241 hours


Is this correct?
(It took me about 3 hours to get this)

 

[SammyS]

I eventually did this by altering this equation:

N = N₀e^-λt

if one divides by N₀ then the left hand side is just a faction of the start and end amounts (whether this is in terms of amount of atoms or in terms of mass it will be the same.

In this case the left hand side of the equation (the fraction) will be 1/5 as 1 is 1/5 of 5.


This gives me 1/5 = e^-λt

taking ln of both sides to eliminate e gives:

ln(1/5) = -λt

t = ln(1/5)/-λ

I substitute the value of decay constant (λ) to get my overall answer of 543.241 hours

Is this correct?
(It took me about 3 hours to get this)

Yes, that's correct.

Without rounding until the final answer, I get t_1/5 = 543.331 hours .

 

[Nickie]

To calculate the time taken for an element to decay from one mass to another, you can use the decay constant and the half-life of the element. The decay constant, denoted by λ, represents the rate at which the element decays. It is equal to ln(2)/t1/2, where t1/2 is the half-life of the element.

In this case, the decay constant for 24Na is 8.23x10^-7. Now, to determine the time taken for the sample to contain 1μg of undecayed atoms, we can use the following equation:

N(t) = N0e^-λt

Where N(t) is the amount of undecayed atoms at time t, N0 is the initial amount of undecayed atoms, and e is the base of the natural logarithm.

Plugging in the values, we get:

1μg = 5μg * e^-8.23x10^-7 * t

Solving for t, we get:

t = ln(1/5) / -8.23x10^-7 ≈ 265.5 hours

Therefore, it would take approximately 265.5 hours for a 5μg sample of 24Na to contain 1μg of undecayed atoms.