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Use the graphs of the hydrogen 2s and 2p orbitals (probability density) to explain why the energy of the 2s orbital is lower even though it has a higher expectation value for the radius

Use the graphs of the hydrogen 2s and 2p orbitals (probability density) to explain why the energy of the 2s orbital is lower even though it has a higher expectation value for the radius

Chemistry
Chemistry
10th Edition
ISBN:9781305957404
Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Chapter1: Chemical Foundations
Section: Chapter Questions
Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...
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The image contains two graphs which depict the radial probability distributions for the 2s and 2p atomic orbitals. **Graph 1: 2s Orbital** - **X-axis**: Distance from the nucleus. - **Y-axis**: Probability density. - **Graph Description**: The graph shows a curve that begins at the origin with a small peak close to zero, decreases, and then rises to a larger peak before tapering off as it moves right. This indicates the probability distribution of finding an electron within the 2s orbital. **Graph 2: 2p Orbital** - **X-axis**: Distance from the nucleus. - **Y-axis**: Probability density. - **Graph Description**: The graph shows a single peak that starts from the origin and reaches its maximum value considerably earlier than in the 2s orbital before gradually decreasing. This represents the probability distribution of an electron in the 2p orbital. ### Explanation These graphs illustrate how electrons are distributed in different atomic orbitals. The 2s orbital has two peaks, indicating a region closer to the nucleus with some probability density and another further out where the density is higher. In contrast, the 2p orbital has a single peak, showing a different distribution pattern with no initial small peak. Understanding these distributions is crucial for comprehending electron configurations and the chemical behavior of elements.
Transcribed Image Text:The image contains two graphs which depict the radial probability distributions for the 2s and 2p atomic orbitals. **Graph 1: 2s Orbital** - **X-axis**: Distance from the nucleus. - **Y-axis**: Probability density. - **Graph Description**: The graph shows a curve that begins at the origin with a small peak close to zero, decreases, and then rises to a larger peak before tapering off as it moves right. This indicates the probability distribution of finding an electron within the 2s orbital. **Graph 2: 2p Orbital** - **X-axis**: Distance from the nucleus. - **Y-axis**: Probability density. - **Graph Description**: The graph shows a single peak that starts from the origin and reaches its maximum value considerably earlier than in the 2s orbital before gradually decreasing. This represents the probability distribution of an electron in the 2p orbital. ### Explanation These graphs illustrate how electrons are distributed in different atomic orbitals. The 2s orbital has two peaks, indicating a region closer to the nucleus with some probability density and another further out where the density is higher. In contrast, the 2p orbital has a single peak, showing a different distribution pattern with no initial small peak. Understanding these distributions is crucial for comprehending electron configurations and the chemical behavior of elements.
**Transcription and Explanation for Educational Website** **Text:** "Use the graphs of the hydrogen 2s and 2p orbitals (probability density) to explain why the energy of the 2s orbital is lower even though it has a higher expectation value for the radius." **Explanation:** In understanding atomic orbitals, it's essential to consider both the probability density and the energy levels associated with electrons in various orbitals. Specifically, examining hydrogen's 2s and 2p orbitals reveals interesting insights into how energy levels and expected radii can differ. 1. **Probability Density Graphs**: - The 2s orbital shows a spherical distribution of probability density with nodes where the probability is zero. This spherical nature is crucial since it allows some electron density to be closer to the nucleus. - The 2p orbital, on the other hand, has a dumbbell shape with a higher concentration directly further from the nucleus, reducing near-nuclear penetration. 2. **Energy Considerations**: - Despite having a higher expectation value for the radius, the 2s orbital is lower in energy compared to the 2p orbital. This is because of the 2s orbital's penetration: significant probability density near the nucleus resulting in stronger nuclear attraction. - The 2p orbital, by contrast, experiences less effective nuclear charge due to its shape, leading to a higher energy level. By analyzing these orbitals' probability distributions, one can appreciate the complex interplay between electron distribution and energy levels that defines atomic behavior.
Transcribed Image Text:**Transcription and Explanation for Educational Website** **Text:** "Use the graphs of the hydrogen 2s and 2p orbitals (probability density) to explain why the energy of the 2s orbital is lower even though it has a higher expectation value for the radius." **Explanation:** In understanding atomic orbitals, it's essential to consider both the probability density and the energy levels associated with electrons in various orbitals. Specifically, examining hydrogen's 2s and 2p orbitals reveals interesting insights into how energy levels and expected radii can differ. 1. **Probability Density Graphs**: - The 2s orbital shows a spherical distribution of probability density with nodes where the probability is zero. This spherical nature is crucial since it allows some electron density to be closer to the nucleus. - The 2p orbital, on the other hand, has a dumbbell shape with a higher concentration directly further from the nucleus, reducing near-nuclear penetration. 2. **Energy Considerations**: - Despite having a higher expectation value for the radius, the 2s orbital is lower in energy compared to the 2p orbital. This is because of the 2s orbital's penetration: significant probability density near the nucleus resulting in stronger nuclear attraction. - The 2p orbital, by contrast, experiences less effective nuclear charge due to its shape, leading to a higher energy level. By analyzing these orbitals' probability distributions, one can appreciate the complex interplay between electron distribution and energy levels that defines atomic behavior.
Expert Solution
Step 1

S orbital is small in size and effective nuclear charge is greater and hence it experience more effective nuclear charge than p orbital and is more stable and hence has less energy . The orbital which have less energy is always more stable as compared to orbital having more energy. 
hence we can conclude that energy is inversely proportional to stability. 

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