Question

The barium isotope 133ba has a half-life of 10.5 years. a sample begins with 1.1×1010 133ba atoms. how many are left after (a) 5 years, (b) 30 years, and (c) 190 years?

Answer 1

Answer: Atoms of
`_(56)^(133)\textrm{Ba}` left in

a)
`N=78.608* 10^8atoms`

b)
`N=14.650* 10^8atoms`

c)
`N=31.35* 10^3atoms`

Explanation: The given reaction is a type of radioactive decay and all the radioactive decay follows first order reactions. Hence, to calculate the rate constant, we use the formula:

`k=(0.693)/(t_(1/2))`

`t_(1/2)=10.5years`

`k=(0.693)/(10.3years)\\k=0.0672years^(-1)`

To calculate how much amount of sample is left, we use the rate law expression for first order kinetics, which is:

`N=N_oe^(-kt)` ....(1)

where,

k = rate constant =
`0.0672years^(-1)`

t = time taken for decay process

`N_o` = initial amount of the reactant =
`1.1* 10^(10)g`

N = amount left after decay process

• For a)

t = 5 years

Putting values in equation 1, we get:

`N=(1.1* 10^(10))* e^{(-0.0672years^(-1)* 5years)}\\N=78.608* 10^8atoms`

• For b)

t = 30 years

Putting values in equation 1, we get:

`N=(1.1* 10^(10))* e^{(-0.0672years^(-1)* 30years)}\\N=14.650* 10^8atoms`

• For c)

t = 190 years

Putting values in equation 1, we get:

`N=(1.1* 10^(10))* e^{(-0.0672years^(-1)* 190years)}\\N=31.35* 10^3atoms`

Answer 2

a) 7.9x10^9 b) 1.5x10^9 c) 3.9x10^4 To determine what percentage of an isotope remains after a given length of time, you can use the formula p = 2^(-x) where p = percentage remaining x = number of half lives expired. The number of half lives expired is simply x = t/h where x = number of half lives expired t = time spent h = length of half life. So the overall formula becomes p = 2^(-t/h) And since we're starting with 1.1x10^10 atoms, we can simply multiply that by the percentage. So, the answers rounding to 2 significant figures are: a) 1.1x10^10 * 2^(-5/10.5) = 1.1x10^10 * 0.718873349 = 7.9x10^9 b) 1.1x10^10 * 2^(-30/10.5) = 1.1x10^10 * 0.138011189 = 1.5x10^9 c) 1.1x10^10 * 2^(-190/10.5) = 1.1x10^10 * 3.57101x10^-6 = 3.9x10^4