The half life of radium is 1590 yrs. how long will it take to 1 gm of element to loose 0.1gm
Steve's calculation is correct. It is worth noting that if you had an actual 1 gram block of Radium-226, its mass would NOT be reduced all the way to 0.9 grams after 241.7 years. Radium-226 decays by alpha emission to Radon-222, which goes quickly through a sequence of further decay ending with stable Lead-206, much of which would still be trapped within the metal block and would contribute to its remaining mass.
To determine the time it takes for 1 gram of radium to lose 0.1 grams, we can use the formula for exponential decay:
N(t) = N(0) * (1/2)^(t/T)
where:
N(t) is the final amount
N(0) is the initial amount
t is the time passed
T is the half-life of the element
Let's solve for t using the given information:
N(t) = N(0) - 0.1 g (since we want to find the time it takes to lose 0.1 grams)
N(0) = 1 g (initial amount)
T = 1590 years (half-life of radium)
Substituting the values into the equation:
1 - 0.1 = 1 * (1/2)^(t/1590)
0.9 = (1/2)^(t/1590)
To solve for t, we can take the logarithm of both sides (base 2):
log2(0.9) = (t/1590) * log2(1/2)
Using the logarithm identity log_a(b) = log_c(b) / log_c(a), where log2(x) = ln(x)/ln(2):
log(0.9) / log(2) = (t/1590) * (ln(1/2)/ln(2))
Simplifying:
log(0.9) / log(2) = -t/1590
Multiplying both sides by -1:
log(0.9) / log(2) = t/1590
Finally, solving for t:
t = (1590 * log(0.9) / log(2))
Using a calculator, we find:
t ≈ -431.445 years
Since time cannot be negative, it seems there may be an error in the calculation or data provided. Please check the given values again or consult a subject matter expert for clarification.
To calculate the time it takes for 1 gram of radium to decay and lose 0.1 grams, we can use the concept of half-life.
The half-life of radium is given as 1590 years. This means that after every 1590 years, half of the radium atoms will decay.
Now, let's break down the problem step by step:
1. Determine the number of half-lives needed:
Since we want to know the time it takes to lose 0.1 grams, we need to find out how many half-lives it would take for the initial 1 gram of radium to reduce to 0.9 grams (1 gram - 0.1 gram).
Using the half-life formula:
Number of half-lives = (ln(N₀/Nf))/ln(2)
Where N₀ is the initial amount of the substance and Nf is the final amount.
Number of half-lives = (ln(1/0.9))/ln(2) ≈ 0.0447
So, approximately 0.0447 half-lives are needed to decay from 1 gram to 0.9 grams.
2. Calculate the time for one half-life:
We know that one half-life of radium is 1590 years.
So, the time for one half-life is 1590 years.
3. Calculate the time for 0.0447 half-lives:
To find the time it takes for 0.0447 half-lives, we multiply the time for one half-life (1590 years) by the decimal fraction of half-lives (0.0447).
Time = 1590 years x 0.0447 ≈ 71.173 years
Therefore, it would take approximately 71.173 years for 1 gram of radium to decay and lose 0.1 grams.
that means that 90% remains, so
(1/2)^(t/1590) = 9/10
t = 241.7 years