Certainly! Let's delve into the difference in binding energy for the 1s electrons in nitrogen and oxygen atoms, using principles from Coulomb's law and atomic structure.
### Coulomb's Law and Atomic Structure:
1. Atomic Nucleus Charge:
- Nitrogen (N) has 7 protons in its nucleus.
- Oxygen (O) has 8 protons in its nucleus.
2. Coulomb’s Law:
- Coulomb's law governs the electrostatic interaction between charged particles. The force [tex]\( F \)[/tex] between two charges [tex]\( q_1 \)[/tex] and [tex]\( q_2 \)[/tex] separated by a distance [tex]\( r \)[/tex] is given by:
[tex]\[
F = k_e \frac{q_1 q_2}{r^2}
\][/tex]
where [tex]\( k_e \)[/tex] is Coulomb’s constant.
- The potential energy [tex]\( U \)[/tex] associated with these charges is:
[tex]\[
U = k_e \frac{q_1 q_2}{r}
\][/tex]
- In the context of a hydrogen-like atom (one-electron system), the binding energy of an electron is directly related to the electrostatic attraction between the negatively charged electron and the positively charged nucleus.
3. Comparing Nitrogen and Oxygen:
- Both nitrogen and oxygen bind their 1s electrons through electrostatic interactions between the protons in the nucleus and the 1s electrons.
- Nitrogen (N), with its 7 protons, exerts an electrostatic force on the 1s electrons that is proportional to the nuclear charge.
- Oxygen (O), having 8 protons, has a higher nuclear charge compared to nitrogen.
4. Implication of Higher Nuclear Charge:
- The higher nuclear charge in oxygen means a stronger electrostatic attraction between the nucleus and the 1s electrons.
- A stronger attraction translates to higher binding energy since more energy is required to overcome this attraction and remove the electron.
### Binding Energy Values:
- The binding energy for the 1s electrons in nitrogen is [tex]\( 39.6 \)[/tex] MJ/mol.
- The binding energy for the 1s electrons in oxygen is [tex]\( 52.6 \)[/tex] MJ/mol.
### Explanation of Difference:
Given that the binding energy for 1s electrons in oxygen ([tex]\( 52.6 \)[/tex] MJ/mol) is higher than that for nitrogen ([tex]\( 39.6 \)[/tex] MJ/mol), the difference in binding energy can be calculated as follows:
[tex]\[
\text{Difference in binding energy} = \text{Binding energy of oxygen} - \text{Binding energy of nitrogen}
\][/tex]
[tex]\[
\text{Difference in binding energy} = 52.6 \, \text{MJ/mol} - 39.6 \, \text{MJ/mol} = 13.0 \, \text{MJ/mol}
\][/tex]
### Conclusion:
Therefore, the difference in binding energy of [tex]\( 13.0 \)[/tex] MJ/mol between the 1s electrons of nitrogen and oxygen can be explained by the greater nuclear charge of oxygen. This greater charge increases the electrostatic attraction between the nucleus and the 1s electrons, leading to a higher binding energy for oxygen compared to nitrogen.
