The carbon−14 decay rate of a sample obtained from a young tree is 0.266 disintegration per second per gram of the sample. another wood sample prepared from an object recovered at an archaeological excavation gives a decay rate of 0.178 disintegration per second per gram of the sample. what is the age of the object? (the half-life of carbon−14 is 5715 years.) × 10 years enter your answer in scientific notation.

Explanation:

1) Carbon-14 disintegration rate of a young (alive) tree: 0.266 per second per gram.

2) Carbon-14 disintegration rate of a wood sample prepeared from an object recovered at an archaelogical excavation (dead matter): 0.178 per second per gram.

3) Ratio of decay: A/A₀ = 0.178 / 0.266 = 0.669

4) Half-life equation: A/A₀ = (1/2)ⁿ, where n is the number of half-lives since the object died.

5) Substitute and solve for n:

That means that 0.578 half-life has elapsed since the wood with which the object was created was dead matter.

6) Convert number of half-lives to years:

part 1: the half-life of the element is 22 years.

part 2: it will take 132.02 year for the sample to decay from 308.0 g to 4.8125 g.

explanation:

part 1: there are two ways to determine the half life time.

first way: the half life time can be determined directly from its definition as it is the time required for the sample to decay to its half concentration.

from the data the initial concentration was 45.0 g and decayed to its half value (22.5 g) in 22.0 years.

second way: the radio active decay is considered first order reaction so we can use the laws of first order reactions.

k = (1/t) ln (a/a-x), where k is the rate constant of the reaction,

t is the time of the reaction

a is the initial concentration of the reactant

a-x is the remaining concentration at time (t)

a = 45.0 g the initial concentration at t = 0 year.

at t = 22 year, a-x = 22.5 g

then k = (1/22 year) ln (45.0 g/ 22.5 g) = 0.0315 year-1.

the half-life time (t1/2) = ln 2/ k = 0.693/ k =0.693/ 0.0315 = 22 years.

part 2:

using the same law k = (1/t) ln (a/a-x),

using the given data; a = 308.0 g and a-x = 4.8125 g and the calculated k from the part 1 (k = 0.0315 year-1)

t = (1/k) ln (a/a-x) = (1/0.0315) ln (308.0/ 4.8125) = 132.02 years