Final answer:
The instantaneous rate of production of nitrogen atoms from a sample with a carbon-14 content of 6.5 x 10^-9 M can be found using the first-order decay equation d[N]/dt = k [C], where k is the decay constant and [C] is the concentration of carbon-14. Substituting the values gives an instantaneous rate of d[N]/dt = 7.865 x 10^-13 M/year.
Step-by-step explanation:
The question pertains to the rate of neutron production equation in the context of radioactive decay, specifically regarding carbon-14 (14C). The rate constant provided for the decay of 14C is 1.21 x 10-4 year-1. During the decay process, 14C is transformed into nitrogen atoms and beta particles (electrons). The query asks for the instantaneous rate of production of nitrogen (N) atoms in a sample with a carbon-14 content of 6.5 x 10-9 M (molar).
To solve this, one can use the first-order decay equation for rate of change, which is given by the formula:
d[N]/dt = k [C]
Where:
- d[N]/dt is the instantaneous rate of production of nitrogen atoms,
- k is the rate constant of carbon-14 decay,
- [C] is the concentration of carbon-14.
Substituting the rate constant and the 14C concentration into the equation:
d[N]/dt = (1.21 x 10-4 year-1) (6.5 x 10-9 M)
The calculation yields:
d[N]/dt = 7.865 x 10-13 M/year
Which represents the instantaneous rate at which nitrogen atoms are produced from the decay of carbon-14 in the given sample.