Question 34 of 62intermediate🔧 ApplyNumerical3 marks

Calculate the energy required for the process He⁺ (g) → He²⁺ (g) + e⁻ The ionization energy for the H atom in the ground state is 2.18×102.18 \times 10⁻¹⁸ J atom⁻¹

Correct Answer

The energy required for the process is 8.72×10−188.72 \times 10^{-18} J atom−1^{-1}.

Exercise: EXERCISES | Q: 2.34 | (Page 41)
For More Understanding

Explanation

The energy of a stationary state for a hydrogen-like species is given by En=−(2.18×10−18 J)Z2n2 atom−1E_n = -\frac{(2.18 \times 10^{-18} \text{ J}) Z^2}{n^2} \text{ atom}^{-1}. For a He+^+ ion, the atomic number Z=2Z = 2. The given process represents the ionization of He+^+ from its ground state (n=1n = 1) to an ionized state (n=∞n = \infty). The energy at n=∞n = \infty is zero. Therefore, the energy required is the difference between the energy at n=∞n = \infty and the energy at n=1n = 1.

Solution Steps

  1. Step 1: Write the formula for the energy of a stationary state for a hydrogen-like species: En=−(2.18×10−18 J)Z2n2 atom−1E_n = -\frac{(2.18 \times 10^{-18} \text{ J}) Z^2}{n^2} \text{ atom}^{-1}.

  2. Step 2: For the He+^+ ion, the atomic number Z=2Z = 2. The ionization process occurs from the ground state, so n=1n = 1. Calculate the ground state energy of He+^+: E1=−(2.18×10−18 J)(22)12=−8.72×10−18E_1 = -\frac{(2.18 \times 10^{-18} \text{ J})(2^2)}{1^2} = -8.72 \times 10^{-18} J.

  3. Step 3: When the electron is completely removed, it reaches the ionized state where n=∞n = \infty and the energy E∞=0E_{\infty} = 0.

  4. Step 4: Calculate the energy required for the process: ΔE=E∞−E1=0−(−8.72×10−18 J)=8.72×10−18\Delta E = E_{\infty} - E_1 = 0 - (-8.72 \times 10^{-18} \text{ J}) = 8.72 \times 10^{-18} J atom−1^{-1}.

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