Question

The energy of the second orbit of a hydrogen atom is \( -5.45 \times 10^{-19} \) J. What is the energy of the first orbit of \( Li^{2+} \) ion (in J)?

• A. \( -1.962 \times 10^{-18} \)
• B. \( -1.962 \times 10^{-17} \)
• C. \( -3.924 \times 10^{-17} \)
• D. \( -3.924 \times 10^{-18} \)

Hint:

The energy levels of multi-electron atoms scale with \( Z^2 \); for \( Li^{2+} \), energy is 9 times that of hydrogen.

Answer 1

The Correct Option is B

Step 1: {Energy Formula}
The electron's orbital energy is calculated using the formula: \[ E_n = -13.6 \frac{Z^2}{n^2} eV \] Step 2: {Energy for \( Li^{2+} \)}
For \( Li^{2+} \), the atomic number \( Z = 3 \). For the first orbit, \( n = 1 \): \[ E_1 = -13.6 \times \frac{3^2}{1^2} eV \] \[ E_1 = -122.4 eV \] Step 3: {Conversion to Joules}
Convert the energy from electronvolts (eV) to Joules (J): \[ E_1 = (-122.4) \times (1.602 \times 10^{-19}) \] \[ E_1 = -1.962 \times 10^{-17} { J} \] The correct answer is (B).