To determine which species among \( O, O^+, O^{2+}, \) and \( O^{2-} \) has the most positive and most negative value of electron gain enthalpy (\( \Delta H_{eg} \)), we need to analyze the electron configurations and the stability of each species.
### Step-by-Step Solution:
1. **Understanding Electron Gain Enthalpy (\( \Delta H_{eg} \))**:
- Electron gain enthalpy is the energy change when an electron is added to a neutral atom or ion. If energy is released during this process, \( \Delta H_{eg} \) is negative, indicating a high electron affinity. Conversely, if energy is required to add an electron, \( \Delta H_{eg} \) is positive, indicating low electron affinity.
2. **Analyzing the Electron Configurations**:
- The electron configuration of neutral oxygen (O) is \( 1s^2 2s^2 2p^4 \). It needs 2 more electrons to achieve a stable octet configuration.
- For \( O^+ \) (oxygen with one less electron), the configuration is \( 1s^2 2s^2 2p^3 \). It has a higher effective nuclear charge compared to neutral oxygen.
- For \( O^{2+} \) (oxygen with two less electrons), the configuration is \( 1s^2 2s^2 2p^2 \). It has an even higher effective nuclear charge and thus a stronger attraction for additional electrons.
- For \( O^{2-} \) (oxygen with two extra electrons), the configuration is \( 1s^2 2s^2 2p^6 \). It has a stable octet, and adding another electron would lead to electron-electron repulsion, making it energetically unfavorable.
3. **Determining the Most Positive \( \Delta H_{eg} \)**:
- Among these species, \( O^{2-} \) has a stable octet. Adding another electron would require energy to overcome the repulsion between electrons, leading to a positive \( \Delta H_{eg} \). Therefore, \( O^{2-} \) has the most positive \( \Delta H_{eg} \).
4. **Determining the Most Negative \( \Delta H_{eg} \)**:
- \( O^{2+} \) has a strong effective nuclear charge due to the loss of two electrons. This results in a high attraction for additional electrons, leading to a highly negative \( \Delta H_{eg} \). Thus, \( O^{2+} \) has the most negative \( \Delta H_{eg} \).
5. **Final Conclusion**:
- The species with the most positive \( \Delta H_{eg} \) is \( O^{2-} \), and the species with the most negative \( \Delta H_{eg} \) is \( O^{2+} \).
### Summary:
- **Most Positive \( \Delta H_{eg} \)**: \( O^{2-} \)
- **Most Negative \( \Delta H_{eg} \)**: \( O^{2+} \)
