Chapter 19, Problem 12STP

(a)

Interpretation Introduction

Interpretation: Balance equation of beta emission of carbon-14, is to be written.

Concept Introduction: Nuclear reactions can be expressed similarly to that of chemical equations. The transformations of particles follow certain conservation laws. In balance a nuclear equation, the atomic numbers and mass number of all species on either side of the arrow must be equal.

The beta particle has one unit negative charge and its emission does not change the mass number.

Answer to Problem 12STP

The balanced nuclear equation is ${}_6^{14}\text{C}\to {}_7^{14}\text{N }+{}_{-1}^{\text{ 0}}e$

Explanation of Solution

Let the atomic numbers of the missing element is $A$ .

The total atomic number on the right side of the equation is $=A-1$

According to the conservation of atomic number,

  $\begin{array}{c}A-1=6\\ A=7\end{array}$

Atomic number 7 corresponds to nitrogen. The balanced nuclear equation is ${}_6^{14}\text{C}\to {}_7^{14}\text{N }+{}_{-1}^{\text{ 0}}e$

(b)

Interpretation Introduction

Interpretation: A balance equation of nitrogen-14 with a neutron in 1:1 molar ratio is to be written.

Concept Introduction: Nuclear reactions can be expressed similarly to that of chemical equations. The transformations of particles follow certain conservation laws. In balance a nuclear equation, the atomic numbers and mass number of all species on either side of the arrow must be equal.

The beta particle has one unit negative charge and its emission does not change the mass number.

Answer to Problem 12STP

The balanced nuclear equation: $\text{N}+n\to C+\text{H}$

Explanation of Solution

The total atomic number on the right side of the equation is $=6+1=7$

The total atomic number on the left side of the equation is $=7+0=7$

The total mass number on the right side of the equation is $=14+1=15$

The total mass number on the left side of the equation is $=14+1=15$

So, both atomic numbers and mass numbers are perfectly balanced in the nuclear equation.

(c)

Interpretation Introduction

Interpretation: Carbon-14 used to know the age of wooden artifacts is to be determined.

Concept Introduction: Carbon-14 dating is a way of determining the objective age of certain carbon-based archeological artifacts of a biological origin up to about 50,000 years old.

Answer to Problem 12STP

The determination of the age of a sample of the wood consists of determining the ratio of amounts $\text{C}$ and $\text{C}$ in both pieces of wood that is that fresh piece and dead piece.

Explanation of Solution

The atmospheric carbon dioxide is a mixture of $\text{C}{\text{O}}_{\text{2}}$ and $\text{C}{\text{O}}_{\text{2}}$ which are present in a fixed ratio. Plants absorb carbon dioxide from the atmosphere and form cellulose (wood). When the plant is alive, the ratio of $C$ and $C$ atoms is the same as that of the atmosphere but when the plant died, this ratio starts decreasing due to a decrease in the amount of $\text{C}$ in the plant. $\text{C}$ decays by beta emission. The equation is: ${}_6^{14}\text{C}\to {}_7^{14}\text{N }+{}_{-1}^{\text{ 0}}e$

Now the age of the rock can be determined by the following equation:

  $t=\frac{t_{1/2}}{0.693}\ln \frac{N_0}{N}$

Where, $t=$ age of the wood.

  $t_{1/2}=$ half-life of $\text{C}$

  $N_0=\frac{\text{C}}{\text{C}}$ ratio in the living (fresh) plant

  $N=\frac{\text{C}}{\text{C}}$ ratio in the dead (cut) plant