## **Question 3: Drawing Orbitals** ### **Step 1: Draw and explain the orbitals’ shapes and axes** #### (i) **3d_z²** - **Shape:** This orbital has a "doughnut" or "dumbbell with a donut" shape. There are two lobes along the z-axis and a torus (donut) of electron density in the xy-plane. - **Axes:** Align the vertical axis as z. - **Phases:** The two lobes along the z-axis have one phase (e.g., +), and the torus in the xy-plane has the opposite phase (−). **Explanation:** The 3d_z² orbital is unique among d orbitals, featuring a donut-shaped electron cloud in the xy-plane and two elongated lobes along the z-axis. --- #### (ii) **2p_x** - **Shape:** This orbital looks like two elongated lobes on either side of the nucleus along the x-axis. - **Axes:** Horizontal axis is x. - **Phases:** One lobe is +, the other is −. **Explanation:** The 2p_x orbital is a simple dumbbell shape aligned along the x-axis, with one lobe having a positive phase and the other a negative phase. --- #### (iii) **4d_xz** - **Shape:** Four lobes oriented between the x and z axes (in the xz-plane, at 45° to both). - **Axes:** x and z axes. - **Phases:** Opposite lobes have the same phase; adjacent lobes have opposite phases (+ and − alternately). **Explanation:** The 4d_xz orbital consists of four lobes, lying between the x and z axes, characteristic of d-type orbitals, with alternating phases. --- ### **Step 2: Final Answer and Diagram Summary** - **3d_z²:** Two elongated lobes along z (both +), donut in xy-plane (−). - **2p_x:** Two lobes along x-axis (left lobe +, right lobe −). - **4d_xz:** Four lobes in xz-plane, alternating + and −. **Final Answer:** These are the correct three-dimensional representations of the requested orbitals, with axes and phases indicated as per quantum mechanical convention. --- ## **Question #4: Bohr Energy Levels for Boron** ### **Step 1: Bohr Equation and Energy Calculation** The Bohr equation for energy levels: \[ E_n = -\frac{13.6\,\text{eV}}{n^2} Z^2 \] For boron, Z = 5. So, \[ E_n = -13.6 \frac{5^2}{n^2} = -340 \frac{1}{n^2} \text{ eV} \] Calculate for n = 1, 2, 3, 4, ∞: - n = 1: \( E_1 = -340\, \text{eV} \) - n = 2: \( E_2 = -85\, \text{eV} \) - n = 3: \( E_3 = -37.8\, \text{eV} \) - n = 4: \( E_4 = -21.25\, \text{eV} \) - n = ∞: \( E_\infty = \, \text{eV} \) **Explanation:** These are the energy levels for a hydrogen-like boron atom (single electron, ignoring electron-electron interactions). --- ### **Step 2: Energy Level Diagram and Electron Configuration** Boron has 5 electrons. Fill orbitals using the aufbau principle: - 1s² (2 electrons) - 2s² (2 electrons) - 2p¹ (1 electron) **Diagram Layout:** - n=1: 1s (2 electrons) - n=2: 2s (2 electrons), 2p (1 electron) - n=3: 3s, 3p, 3d (empty) - n=4: 4s, 4p, 4d, 4f (empty) - n=∞: Ionization limit **Final Answer:** The energy level diagram for boron, using the Bohr equation, shows electrons filling up to 2p. Energy levels for n = 1 (−340 eV), n = 2 (−85 eV), n = 3 (−37.8 eV), n = 4 (−21.25 eV), and n = ∞ ( eV). Each shell and subshell is labeled, and electrons are placed as per boron’s electron configuration. --- If you need actual orbital diagrams or a sketched energy level diagram, let me know!

## **Question 3: Drawing Orbitals** ### **Step 1: Draw and explain the orbitals’ shapes and axes** #### (i) **3d_z²** - **Shape:** This orbital has a "doughnut" shape with two lobes along the z-axis and a torus in the xy-plane. - **Axes:** The vertical axis is z. - **Phases:** The two lobes along the z-axis are positive (+) and the torus in the xy-plane is negative (−). **Explanation:** The 3d_z² orbital features a unique shape with a toroidal electron density in the xy-plane and two lobes extending along the z-axis, where one phase is positive and the other is negative. --- #### (ii) **2p_x** - **Shape:** This orbital resembles two lobes positioned on either side of the nucleus along the x-axis. - **Axes:** The horizontal axis is x. - **Phases:** One lobe is positive (+) and the other is negative (−). **Explanation:** The 2p_x orbital is a dumbbell shape oriented along the x-axis, indicating the different phases of the lobes. --- #### (iii) **4d_xy** - **Shape:** This orbital has four lobes oriented in the xy-plane, between the x and y axes. - **Axes:** The x and y axes. - **Phases:** Opposite lobes have the same phase (+), and adjacent lobes have opposite phases (−). **Explanation:** The 4d_xy orbital consists of four lobes positioned in the xy-plane, characteristic of d-type orbitals, with alternating phases indicated. --- ### **Step 2: Final Answer and Diagram Summary** - **3d_z²:** Two elongated lobes along z (both +), donut in the xy-plane (−). - **2p_x:** Two lobes along the x-axis (left lobe +, right lobe −). - **4d_xy:** Four lobes in the xy-plane, alternating + and −. **Final Answer:** These representations of the specified orbitals are accurate, with the axes and phases clearly defined according to quantum mechanical conventions. --- ## **Question #4: Bohr Energy Levels for Boron** ### **Step 1: Bohr Equation and Energy Calculation** The Bohr equation for energy levels is given by: \[ E_n = -\frac{13.6\,\text{eV}}{n^2} Z^2 \] For boron (Z = 5), the equation becomes: \[ E_n = -\frac{13.6 \cdot 5^2}{n^2} = -\frac{340}{n^2} \text{ eV} \] Calculate for n = 1, 2, 3, 4, and ∞: - \( E_1 = -340\, \text{eV} \) - \( E_2 = -85\, \text{eV} \) - \( E_3 = -37.78\, \text{eV} \) - \( E_4 = -21.25\, \text{eV} \) - \( E_{\infty} = 0\, \text{eV} \) **Explanation:** These calculations yield the energy levels for a hydrogen-like boron atom, assuming a single electron, neglecting electron-electron interactions. --- ### **Step 2: Energy Level Diagram and Electron Configuration** Boron has 5 electrons, filled following the Aufbau principle: - 1s² (2 electrons) - 2s² (2 electrons) - 2p¹ (1 electron) **Diagram Layout:** - n=1: 1s (2 electrons) - n=2: 2s (2 electrons), 2p (1 electron) - n=3: 3s, 3p, 3d (empty) - n=4: 4s, 4p, 4d, 4f (empty) **Final Answer:** The energy level diagram for boron illustrates the electrons filling up to the 2p subshell, with energy levels calculated for n = 1 (−340 eV), n = 2 (−85 eV), n = 3 (−37.78 eV), n = 4 (−21.25 eV), and n = ∞ (0 eV). Each shell and subshell is labeled, and electrons are placed according to boron’s electron configuration.