The half – life for the disintegration of radium is 1590 years. Calculate the rate constant
in seconds and also how many years will be taken for the disintegration of 80 percent of the material?
Answer
To calculate the rate constant, we can use the formula for radioactive decay:
N(t) = N(0) * e^(-kt)
Where:
N(t) = the amount of material remaining at time t
N(0) = the initial amount of material
k = rate constant
t = time
The half-life (t1/2) is defined as the time it takes for half of the material to decay.
Given that t1/2 for radium is 1590 years, we can find the rate constant (k) using the formula:
t1/2 = (ln(2)) / k
Rearranging the formula, we have:
k = (ln(2)) / t1/2
Substituting the given half-life:
k = (ln(2)) / 1590 ≈ 0.0004359 per year
To convert the rate constant to seconds, we need to multiply the value by the number of seconds in a year. Assuming there are 365.25 days in a year:
k (seconds) = 0.0004359 per year * 365.25 days/year * 24 hours/day * 60 minutes/hour * 60 seconds/minute ≈ 1.377 x 10^(-8) per second
To find out how many years it will take for 80% of the material to decay, we use the equation:
ln(N(t) / N(0)) = -kt
Substituting N(t) = 0.8N(0) (80% remaining), we can solve for t:
ln(0.8) = -k*t
t = ln(0.8) / (-k)
Substituting the calculated value of k:
t = ln(0.8) / (-1.377 x 10^(-8)) ≈ 15404.39 seconds
To convert seconds to years, we divide the value by the number of seconds in a year:
t (years) = 15404.39 seconds / (365.25 days/year * 24 hours/day * 60 minutes/hour * 60 seconds/minute) ≈ 0.487 years
Therefore, approximately 0.487 years will be taken for 80 percent of the radium to disintegrate.
To calculate the rate constant for the disintegration of radium, we can use the formula:
rate constant (k) = (0.693 / half-life)
In this case, the half-life of radium is given as 1590 years. Therefore, we can calculate the rate constant as:
k = 0.693 / 1590
To find the rate constant in seconds, we need to convert the half-life from years to seconds. Since 1 year is approximately equal to 3.1536 × 10^7 seconds, we can convert the half-life as follows:
half-life (in seconds) = half-life (in years) × 3.1536 × 10^7
Now we can substitute the half-life in seconds into the rate constant formula:
rate constant (in seconds) = 0.693 / (half-life (in seconds))
Finally, to calculate the time taken for the disintegration of 80 percent of the material, we can use the equation:
t = -(ln(1 - 0.8) / k)
Where:
t = time taken for disintegration
ln = natural logarithm
k = rate constant
Now let's perform the calculations:
Step 1: Calculate the rate constant in seconds
half-life (in seconds) = 1590 years × 3.1536 × 10^7 seconds/year
= 5.0209 × 10^10 seconds
rate constant (in seconds) = 0.693 / (5.0209 × 10^10 seconds)
≈ 1.38 × 10^-11 s^-1
Step 2: Calculate the time taken for the disintegration of 80 percent of the material
t = -(ln(1 - 0.8) / k)
= -(ln(0.2) / (1.38 × 10^-11 s^-1))
≈ 9.05 × 10^10 seconds
To convert this time into years, divide the value by the number of seconds in a year:
9.05 × 10^10 seconds ÷ 3.1536 × 10^7 seconds/year ≈ 2870 years
Therefore, it will take approximately 2870 years for 80 percent of the radium material to disintegrate.
k = 0.693/t1/2</sub) gives k in years. You can convert that to seconds. I would convert 1590 years to seconds and plug that in to calculate rate in seconds. But use the year calculation above to solve for years. If 80% disintergration then 20% is eft.
ln(No/N) = kt
No = 100
N = 20
k from above in years
t is the unknown. Solve for that in years.