Question

The rate of radioactive decay is first-order, so we can use our knowledge of first-order kinetics to study it. The half-life for the decay of fermium −252 is about 1 day. The half-life for the decay of ruthenium −106 is about 1 year. Which of the following statements is the best conclusion from this information?
A.The rate constant for the decay of fermium -252 is equal to the rate constant for the decay of ruthenium-106.
B.The rate constant for the decay of fermium -252 is less to the rate constant for the decay of ruthenium- 106 .
C.The rate constant for the decay of fermium -252 is greater than the rate constant for the decay of ruthenium-106.
D.It is not possible to make any comparison of the rate constant for the decay of these two isotopes.

Answer 1

Hint: The proportionality constant, which describes the link between the molar concentration of the reactants and the pace of a chemical reaction, is known as the rate constant. The rate constant, also known as the response rate constant or reaction rate coefficient, is indicated by the letter k.

Complete answer: The process by which an unstable atomic nucleus loses energy through radiation is known as radioactive decay (also known as nuclear decay, radioactivity, radioactive disintegration, or nuclear disintegration). The term "radioactive" refers to a substance that contains unstable nuclei. An order of chemical reaction in which the rate of the reaction is proportional to the amount of one reactant and is determined by its concentration. It may be expressed using the formula rate = kA, where k is the reaction rate constant and A is the reactant concentration.
The half-life (symbol ${t_{1/2}}$) is the amount of time it takes for a quantity to decline to half its original value. In nuclear physics, the word is used to explain how rapidly unstable atoms disintegrate radioactively and how long stable atoms survive. The phrase is also used to describe any form of exponential or non-exponential decay in general. The biological half-life of medications and other compounds in the human body, for example, is discussed in medical research. Doubling time is the inverse of half-life.
Now for the given first order reaction,
${{\text{t}}_{1/2}} = \dfrac{{0.693}}{{\;{\text{K}}}}$
We calculate for Fermium
$\left( {{{\text{t}}_{1/2}}} \right) = \dfrac{{{\mathbf{0}}.{\mathbf{693}}}}{{{{\mathbf{K}}_{{\text{fermium }}}}}}$
$\therefore {{\text{K}}_{{\text{fermium }}}} = \dfrac{{{\mathbf{0}}.{\mathbf{693}}}}{1}{\text{ day}}{{\text{ }}^{ - 1}}$
$ \Rightarrow {{\text{K}}_{{\text{fermium }}}} = 0.693{\text{ day}}{{\text{ }}^{ - 1}}$
Now when we calculate for Ruthenium
${\left( {{{\text{t}}_{1/2}}} \right)_{{\text{ruthenium }}}} = \dfrac{{{\mathbf{0}}.{\mathbf{693}}}}{{{{\mathbf{K}}_{{\text{ruthenium }}}}}}$
Using the given data
$1 \times 365{\text{ days }} = \dfrac{{{\mathbf{0}}.{\mathbf{693}}}}{{{{\mathbf{K}}_{{\text{ruthenium }}}}}}$
${{\mathbf{K}}_{{\text{ruthenium }}}} = \dfrac{{{\mathbf{0}}.{\mathbf{693}}}}{{{\mathbf{365}}}}$
$ \Rightarrow {{\mathbf{K}}_{{\text{ruthenium }}}} = 1.9 \times {10^{ - 3}}{\text{ day}}{{\text{ }}^{ - 1}}$
Hence one can conclude that
$\therefore {{\mathbf{K}}_{{\text{fermium }}}} > {{\mathbf{K}}_{{\text{ruthenium }}}}$
Hence option C is correct.

Note:
The decay of discrete things, such as radioactive atoms, is commonly described by a half-life. In such a scenario, the concept that "half-life is the time necessary for exactly half of the entities to decay" does not apply. If just one radioactive atom exists and its half-life is one second, there will be no "half of an atom" remaining after one second.