Question

If the radius of the first orbit of the hydrogen-like ion is \(1.763 \times 10^{-2}\) nm, the energy associated with that orbit (in J) is:

• A. \(+1.962 \times 10^{-17}\)
• B. \(-1.962 \times 10^{-17}\)
• C. \(-0.872 \times 10^{-17}\)
• D. \(-2.18 \times 10^{-18}\)

Hint:

The energy of the electron in hydrogen-like ions is negative and is calculated based on the nuclear charge \(Z\). The formula \(E_n = - \frac{13.6 Z^2}{n^2}\) helps compute energy levels for different ions.

Answer 1

The Correct Option is B

Step 1: Energy Formula for Hydrogen-like Ion The energy of an electron in a hydrogen-like ion (ionized atom) is given by the formula for the energy levels of hydrogen: \[ E_n = - \frac{13.6 \, \text{eV}}{n^2} \] where: - \(n\) is the principal quantum number (for the first orbit, \(n = 1\)), - \(13.6 \, \text{eV}\) is the energy for the first orbit in the hydrogen atom. 

Step 2: Convert Energy from eV to Joules To convert the energy from electron-volts (eV) to joules (J), we use the conversion factor: \[ 1 \, \text{eV} = 1.602 \times 10^{-19} \, \text{J} \] Thus, the energy for the first orbit is: \[ E_1 = - \frac{13.6 \, \text{eV}}{1^2} = -13.6 \, \text{eV} \] \[ E_1 = -13.6 \times 1.602 \times 10^{-19} \, \text{J} = -2.179 \times 10^{-18} \, \text{J} \] 

Step 3: Adjust for Hydrogen-like Ion Energy For a hydrogen-like ion, the energy formula changes based on the ion's nuclear charge \(Z\), with the energy being: \[ E_n = - \frac{13.6 \, Z^2 \, \text{eV}}{n^2} \] In this case, the problem provides the radius, and we must use it to determine \(Z\). The first orbit's radius \(r\) is related to \(n\) and \(Z\) through: \[ r_1 = \frac{0.529 \, \text{Å}}{Z} \] Where: - \(0.529 \, \text{Å}\) is the Bohr radius (the radius of the first orbit in hydrogen). Given the radius \(r_1 = 1.763 \times 10^{-2} \, \text{nm} = 1.763 \times 10^{-1} \, \text{Å}\), we can solve for \(Z\): \[ Z = \frac{0.529}{1.763 \times 10^{-1}} = 3 \] Thus, \(Z = 3\). 

Step 4: Calculate the Energy for the Hydrogen-like Ion Now, we can calculate the energy for the hydrogen-like ion using \(Z = 3\): \[ E_n = - \frac{13.6 \times 9}{1^2} = -122.4 \, \text{eV} \] Convert this energy to joules: \[ E_n = -122.4 \times 1.602 \times 10^{-19} = -1.962 \times 10^{-17} \, \text{J} \] Thus, the energy associated with the first orbit is: \[ \boxed{-1.962 \times 10^{-17} \, \text{J}} \]