24.128 CP - Chemistry: Molecular Nature Of Matter And Change

Question

The reaction that allows for radiocarbon dating is the continual formation of carbon-14 in the upper atmosphere: 

${}_7{}^{14}N{+}_0^{1}n{\to }_6^{14}C{+}_1^{1}H$

What is the energy change associated with this process in eV/reaction and in kJ/mol reaction? (Masses of atoms: ${}_7{}^{14}N=14.003074$amu; ${}_6{}^{14}C=14.003241$ amu;${}_1{}^{1}H=1.007825$  amu;  ${}_0{}^{1}n=1.008665$amu.)

 

Step-by-Step Solution

Answer

$Energy = 60478572.64\frac{k }{mol}$

Step 1: Definition of Continual formation

The carbon cycle depicts the continuous movement of carbon atoms from the atmosphere to the Earth and back to the atmosphere. The amount of carbon in this system does not fluctuate because our planet and its atmosphere are a closed system. The location of carbon in the atmosphere and on Earth is continually changing.

Step 2: Find the energy change associated with this process

We find the mass that is equivalent to this energy from Einstein's equation:

E = mc²

where 4m is the mass difference between reactants and products:

$4m = m_{products}-m_{reac\tan ts}$

$4m = (14.003074+1.008665)-(14.003241+1.007825)$

$4m = 0.000673amu$

Binding energies are commonly expressed in millions of electron volts, that is, in mega-electron volts (MeV):

$1MeV = {10}^{6}eV = 1.602x{10}^{-13}J$

A particularly useful factor converts the atomic mass unit to its energy equivalent in electron volts:

$$\begin{gathered} 1 amu = 931.5x{10}^6 eV = 931.5 MeV \\ Energy = 0.000673amu\left(\frac{931.5MeV}{1amu}\right)\left(\frac{{10}^{6}eV}{1MeV}=6.2690x{10}^{5}eV\right) \\ Energy = 6.2690x{10}^{5}eV\left(\frac{1.602x{10}^{-19} J}{1eV}\right)\left(\frac{1 kJ}{1000J}\right)\left(\frac{6.022x{10}^{23}}{1mol}\right) \\ Energy= 60478572.64\frac{k }{mol} \end{gathered}$$

Thus, $Energy= 60478572.64\frac{k }{mol}$