Q. 5.55 - An Introduction to General, Organic, and Biological Chemistry

Question

Carbon dating of small bits of charcoal used in cave paintings has determined that some of the paintings are from $10000$ to $30000$ old. Carbon-$14$ has a half-life of $5730$. In a $1-\mu g$ sample of carbon from a live tree, the activity of carbon-$14$ is $6.4\mu Ci$. If researchers determine that $1\mu g$ of charcoal from a prehistoric cave painting in France has an activity of $0.80\mu Ci$, what is the age of the painting?$(5.4)$

The technique of carbon dating is used to determine the age of ancient cave paintings.

Step-by-Step Solution

Answer

Therefore, The painting art is $22,933.79$$y.$

Step 1: Introduction.

Cave art is a type of parietal art which can be found on walls of caves (which also contains rock art, or engravings).

Step 2: Given data.

To obtain the lag factor, rewrite equation $1$as follows:

$\lambda =\frac{\ln 2}{t_{1/2}}\dots \dots \left(33\right)$

In the given equation, replace $5730$ $y$ at $t_{1/2}$:

$\lambda =\frac{\ln {\displaystyle }2}{5730y}$

$=\frac{0.693}{5730years}$

$=1.209\times {10}^{-4}{y}^{-1}$

Step 3: Explanation.

A substance's halfway line is the amount of time taken to drop to $50$ percent compared amount. Carbon $-14$ half line $\left(t_{1/2}\right)$ is $5730y$. Decay constant is found by reducing in $2$ by $5730$. As a result, the carbon$-14$ decay constant is $1.209\times {10}^{-4}{y}^{-1}$.

Step 4: Introduction.

Charcoal is a light black carbon residue produced from burning wood (or other animal and plant sources) in a reduced environment in order to eliminate all water and volatile components.

Step 5: Given data.

Put value $1.209\times {10}^{-4}{y}^{-1}$ for $\lambda$, $6.4\mu Ci$ to $N°$ and $0.40\mu Ci$ to $N$ in equation$\left(2\right)$ as:

$t=\left(\frac{2.303}{1.209\times {10}^{-4}{y}^{-1}}\right)\left(\log \frac{6.4\mu \mathrm{Ci}}{0.40\mu \mathrm{Cl}}\right)$

$=(1.9048\times {10}^{4}y)(1.204)$

$=22,933.79y$

The painting of the age is $22,933.79y.$

Step 6: Explanation.

Equation is used to compute the age of artwork by substituting the values of the decaying rate and activities at various intervals $\left(2\right)$. Carbon $-14$ activity is used as the beginning activity, and coal activity is used as the exercise over a certain amount of time. The time period of painting is determined as $22,933.79$ using the activity values and decay constant. The era of art is reflected by the historical period.