Question

Element X is a radioactive isotope such that every 51 years, its mass decreases by half. Given that the initial mass of a sample of Element X is 7500 grams, how long would it be until the mass of the sample reached 6900 grams, to the nearest tenth of a year?

Answer 1

It would take approximately 1.5 half-lives or 76.5 years for the mass of the sample to reach 6900 grams.

Step-by-step explanation:

To find out how long it would take for the mass of the sample to reach 6900 grams, we need to determine the number of half-lives it would take for the mass to decrease from 7500 grams to 6900 grams. Since the mass decreases by half every 51 years, we can set up the equation:

7500 * (1/2)n = 6900

Where n is the number of half-lives. Now, let's solve the equation:

7500 * (1/2)n = 6900

(1/2)n = 6900/7500

(1/2)n = 0.92

Taking the logarithm of both sides, we get:

n * log(1/2) = log(0.92)

n = log(0.92) / log(1/2)

n ≈ 1.47

Since we want the answer to the nearest tenth of a year, we can round up to 1.5

Therefore, it would take approximately 1.5 half-lives or 76.5 years for the mass of the sample to reach 6900 grams.

Answer 2

Answer: It will take 6 years for 7500 grams of X to reach 6900 grams.

Step-by-step explanation:

Expression for rate law for first order kinetics is given by:

`t=(2.303)/(k)\log(a)/(a-x)`

where,

k = rate constant

t = age of sample

a = let initial amount of the reactant

a - x = amount left after decay process

a) for completion of half life:

Half life is the amount of time taken by a radioactive material to decay to half of its original value.

`51=(0.69)/(k)`

`k=(0.69)/(51)=0.0135years^(-1)`

b) for 7500 g to reach to 6900 g

`t=(2.303)/(0.0135)\log(7500)/(6900)`

`t=6years`

It will take 6 years for 7500 grams of X to reach 6900 grams.