Question

A carbon sample from the frame of the picture gives $7$ counts of ${C^{14}}$ per minute per gram of carbon. If freshly cut wood gives $15.3$ counts of ${C^{14}}$ per minute, calculate the age of frame. ( ${t_{1/2}}_{}$ of ${C^{14}}$ = $5570years$ )
A. $6286years$
B. $5527years$
C. $5570years$
D. $4570years$

Answer 1

In the above question we have given initial concentration of carbon sample in frame and also the final concentration of carbon sample in the frame. It is a first order reaction and to find the age of frame we have to use the disintegration constant relation for first order disintegration reaction.
Formula used:
First order disintegration constant:
$k = \dfrac{1}{t}\ln \dfrac{{{a_o}}}{{{a_t}}}$
And, $k = \dfrac{{0.69}}{{{t_{1/2}}}}$
Where, $k$ = disintegration constant
$t$ = time period (age)
${a_o}$ = initial concentration
${a_t}$ = final concentration or concentration at time $t$

Complete step by step answer:
In the above problem, radioactive disintegration reaction is taking place. According to the question the A carbon sample from the frame of picture gives $7$ counts of ${C^{14}}$ per minute per gram of carbon. If freshly cut wood gives $15.3$ counts of ${C^{14}}$ per minute that means:
Initial concentration of carbon sample in the frame $({a_o})$ = $15.3{\text{ counts per minute}}$
Final concentration of carbon sample in the frame at time $t$ $({a_t})$ = $7{\text{ counts per minute}}$
Half life of sample $({t_{1/2}})$ = $5570years$
Now by using the formula $k = \dfrac{{0.69}}{{{t_{1/2}}}}$ , we can calculate the value of disintegration constant:
$\ \Rightarrow k = \dfrac{{0.693}}{{5570}} \\\ \Rightarrow k = 0.000124 \\\ \$
Now, substituting the values in the given formula for First order disintegration constant:
$k = \dfrac{1}{t}\ln \dfrac{{{a_o}}}{{{a_t}}}$
And, $k = \dfrac{{0.69}}{{{t_{1/2}}}}$
Where, $k$ = disintegration constant
$t$ = time period (age)
${a_o}$ = initial concentration
${a_t}$ = final concentration or concentration at time $t$
We will get the time $(t)$ that will be the age of the frame as:
$\ \Rightarrow 0.000124 = \dfrac{1}{t}\ln \dfrac{{15.3}}{7} \\\ \Rightarrow t \approx 6286years \\\ \$
Hence, option A is correct.

Note:
Each and every reaction has a different rate constant. Rate constant is generally used for the idea of rate of reaction and the direction of chemical reaction. Rate constant depends on the temperature and molar concentrations of reactants. Every order of reaction has a different relation between rate constant and half life.