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Arrange the following orbitals of H-atom...

Arrange the following orbitals of H-atom in the increasing order of their energy.
`3p_(x),2s,4d_(xy),3s,4p_(z),3p_(y),4s`

A

`2slt3s=3p_(x)=3p_(y)lt4s=4p_(z)=4d_(xy)`

B

`2slt3slt3p_(x)=3p_(y)lt4s=4p_(z)=4d_(xy)`

C

`2slt3slt3p_(x)=3p_(y)lt4s=4p_(z)=4d_(xy)`

D

`2slt3slt3p_(x)=3p_(y)lt4slt4p_(z)lt4d_(xy)`

Text Solution

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To arrange the given orbitals of the hydrogen atom in increasing order of their energy, we will follow these steps: ### Step 1: Identify the quantum numbers Each orbital is characterized by two quantum numbers: the principal quantum number (n) and the angular momentum quantum number (l). The value of l is determined by the type of orbital: - For s orbitals, l = 0 - For p orbitals, l = 1 - For d orbitals, l = 2 ### Step 2: Calculate n + l values We will calculate the n + l value for each orbital: 1. **3p_x**: n = 3, l = 1 → n + l = 3 + 1 = 4 2. **2s**: n = 2, l = 0 → n + l = 2 + 0 = 2 3. **4d_{xy}**: n = 4, l = 2 → n + l = 4 + 2 = 6 4. **3s**: n = 3, l = 0 → n + l = 3 + 0 = 3 5. **4p_{z}**: n = 4, l = 1 → n + l = 4 + 1 = 5 6. **3p_y**: n = 3, l = 1 → n + l = 3 + 1 = 4 7. **4s**: n = 4, l = 0 → n + l = 4 + 0 = 4 ### Step 3: List the n + l values Now, we can summarize the n + l values: - **2s**: n + l = 2 - **3s**: n + l = 3 - **3p_x**: n + l = 4 - **3p_y**: n + l = 4 - **4s**: n + l = 4 - **4p_z**: n + l = 5 - **4d_{xy}**: n + l = 6 ### Step 4: Arrange based on n + l values The orbitals can be arranged in increasing order of their n + l values: 1. **2s** (n + l = 2) 2. **3s** (n + l = 3) 3. **3p_x** and **3p_y** and **4s** (n + l = 4, these are degenerate) 4. **4p_z** (n + l = 5) 5. **4d_{xy}** (n + l = 6) ### Final Arrangement Thus, the increasing order of energy for the given orbitals is: **2s < 3s < 3p_x = 3p_y = 4s < 4p_z < 4d_{xy}**
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