Considering Bohr's atomic model for hydrogen atom: (A) the energy of H atom in ground state is same as energy of mathrmHe^+$\mathrm{He}^+$ ion in its first excited state. (B) the energy of H atom in ground state is same as that for mathrmLi^++$\mathrm{Li}^{++}$ ion in its second excited state. (C) the energy of H atom in its ground state is same as that of mathrmHe^+$\mathrm{He}^+$ ion for its ground state. (D) the energy of mathrmHe^+$\mathrm{He}^+$ ion in its first excited state is same as that for mathrmLi^++$\mathrm{Li}^{++}$ in its ground state Choose the correct answer from the options given below:

### Related Formula E_n = -13.6 fracZ^2n^2mathrm~eV$$E_n = -13.6 \frac{Z^2}{n^2}\mathrm{~eV}$$ where Z$Z$ is the atomic number and n$n$ is the principal quantum number. ### Core Logic Let's check each statement by calculating energy values: - **H atom ground state (Z = 1, n = 1$Z = 1, n = 1$):** E_H = -13.6 frac1^21^2 = -13.6mathrm~eV$$E_H = -13.6 \frac{1^2}{1^2} = -13.6\mathrm{~eV}$$ - **Statement (A):** mathrmHe^+$\mathrm{He}^+$ ion in its first excited state (Z = 2, n = 2$Z = 2, n = 2$): E_mathrmHe^+, n=2 = -13.6 frac2^22^2 = -13.6mathrm~eV$$E_{\mathrm{He}^+, n=2} = -13.6 \frac{2^2}{2^2} = -13.6\mathrm{~eV}$$ This is equal to E_H$E_H$. Thus, **Statement (A) is correct**. - **Statement (B):** mathrmLi^++$\mathrm{Li}^{++}$ ion in its second excited state (Z = 3, n = 3$Z = 3, n = 3$): E_mathrmLi^++, n=3 = -13.6 frac3^23^2 = -13.6mathrm~eV$$E_{\mathrm{Li}^{++}, n=3} = -13.6 \frac{3^2}{3^2} = -13.6\mathrm{~eV}$$ This is equal to E_H$E_H$. Thus, **Statement (B) is correct**. - **Statement (C):** mathrmHe^+$\mathrm{He}^+$ ion in its ground state (Z = 2, n = 1$Z = 2, n = 1$): E_mathrmHe^+, n=1 = -13.6 frac2^21^2 = -54.4mathrm~eV neq E_H$$E_{\mathrm{He}^+, n=1} = -13.6 \frac{2^2}{1^2} = -54.4\mathrm{~eV} \neq E_H$$ Thus, **Statement (C) is incorrect**. - **Statement (D):** mathrmHe^+$\mathrm{He}^+$ in first excited state (Z=2, n=2$Z=2, n=2$) versus mathrmLi^++$\mathrm{Li}^{++}$ in ground state (Z=3, n=1$Z=3, n=1$): E_mathrmLi^++, n=1 = -13.6 frac3^21^2 = -122.4mathrm~eV neq -13.6mathrm~eV$$E_{\mathrm{Li}^{++}, n=1} = -13.6 \frac{3^2}{1^2} = -122.4\mathrm{~eV} \neq -13.6\mathrm{~eV}$$ Thus, **Statement (D) is incorrect**. ### Step 1: Final Conclusion Statements (A) and (B) are the only correct statements. ### Pattern Recognition Bohr energy is proportional to fracZ^2n^2$\frac{Z^2}{n^2}$. If the ratio of fracZn$\frac{Z}{n}$ is the same for two systems, their quantum state energy levels will be identical. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Physics: Atoms