Question

Which is the correct expression for half-life
A. \[{(t)_{1/2}} = \log 2\]
B. \[{(t)_{1/2}} = \lambda /\log 2\]\[\]
C. \[{(t)_{1/2}} = (\lambda /\log 2)2.303\]
D. \[{(t)_{1/2}} = (2.303\log 2)/\lambda \]

Answer 1

Hint: According to this question, we need to find the correct expression for half-life. So first, we need to define the half-life. It is significant that the formula for the half-life of a reaction varies depending on the order of the reaction.

Complete step by step solution:
Another characteristic of each radionuclide is its half-life. Half-life represents the length of time required for half of the radioactive atoms of a particular radionuclide to decay. A good rule of thumb is that, after seven half-lives, one will have less than one percent of the initial amount of radiation. Based on the radionuclide, this process could be quick or time-consuming – radioactive half-lives can vary from milliseconds to hours, days, and then sometimes millions of years.

For a zero-order reaction, the mathematical expression that can be employed to define the half-life is,
\[{t_{1/2}} = {R_0}/2k\]
For a first-order reaction, the half-life is calculated by
\[{t_{1/2}} = 0.693/k\]
For a second-order reaction, the formula for the half-life of the reaction is,
\[1/k{[R]_0}\]
where \[{t_{1/2}}\] is the half-life of the reaction, \[{[R]_0}\] is the initial reactant concentration, and k is the rate constant of the reaction.

For first-order kinetics,
\[k = \dfrac{{2.303}}{t}{\log _{10}}\dfrac{a}{{a - x}}\]
At \[{t_{1/2}}\],
\[k = \dfrac{{2.303}}{{{t_{1/2}}}}{\log _{10}}\dfrac{a}{{a - \dfrac{a}{2}}}\]
\[\therefore {t_{1/2}} = \dfrac{{2.303}}{k}{\log _{10}}2\]

Therefore the correct answer is option D.

Additional Information: The half-life formula is used to find the half-life of a material that is reducing or decaying in quantity. A substance that is decaying has various rates of decay for various amounts of the substance. As the quantity of the substance decreases the rate of decay also slows down, so it is very hard to find the life of a decaying substance. Therefore, the half-life formula is used to offer the right metrics to describe the life of decaying substances.

Note: Students must know the correct half-life expression. So that they can easily find out the correct expression. If they are unable to remember the half-life expression, they are unable to find correct and accurate expressions.