Question

The weight based ratio of ${{U}^{238}}\,and\,P{{b}^{226}}$ in a sample of rock in 4:3. If the half-life of ${{U}^{238}}$is $4.5\times {{10}^{9}}$years then the age of rock is
(A) $9.0\times {{10}^{9}}\,years$
(B) $6.3\times {{10}^{9}}\,years$
(C) $4.5\times {{10}^{9}}\,years$
(D) $3.78\times {{10}^{9}}\,years$

Answer 1

Hint: We know that radioactivity refers to the particles which are emitted from nuclei as a result of nuclear instability. Because the nucleus experiences the intense conflict between the two strongest forces in nature, it should not be surprising that there are many nuclear isotopes which are unstable and emit some kind of radiation. Atoms found in nature are either stable or unstable. An atom is unstable or we can say radioactive if these forces are unbalanced; if the nucleus has an excess of internal energy. Instability of an atom's nucleus may result from an excess of either neutrons or protons. Based on this concept we have to answer this question.

Complete step-by step answer:
It is known to us that the definition of elimination half-life is the length of time required for the concentration of a particular substance (typically a drug) to decrease to half of its starting dose in the body. Half-life is the time it takes for one-half of the atoms of a radioactive material to disintegrate. Scientists can use the half-life of carbon-14 to determine the approximate age of organic objects.
Let us consider that the total amount of ${{U}^{238}}$in the beginning will be 7 units. So, out of 7 units, 3 has decayed to the $P{{b}^{226}}$
Half-life is $4.5\times {{10}^{9}}.$
So, age of rock is $4.5\times {{10}^{9}}\times \dfrac{{{\log }_{e}}\dfrac{7}{4}}{{{\log }_{e}}2}=3.63\times {{10}^{9}}/years$

Hence, the correct answer is Option D.

Note: It should be known to us that the name Half-Life was chosen because it was evocative of the theme, not cliched, and had a corresponding visual symbol $\lambda$, which represents the decay constant in the half-life equation. Half-life (symbol ${{t}_{1/2}}$) is the time required for a quantity to reduce to half of its initial value. The term is commonly used in nuclear physics to describe how quickly unstable atoms undergo, or how long stable atoms survive, radioactive decay.
It should be known to us that many uses of radiation help to ensure the high quality and safety of our daily lives. Smoke detectors to warn us of fire, x-ray machines to detect weapons or other devices in luggage and cargo, and certain types of imaging to look for diseases are all applications of radiation for the benefit of society.