The ionization enthalpy of hydrogen atom is 1.312 x 10^6 J mol-1 | Chemca Solution

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Question

The ionization enthalpy of hydrogen atom is \( 1.312 \times 10^6 \text{ J mol}^{-1} \). The energy required to excite the electron in the atom from \( n = 1 \) to \( n = 2 \) is:

(a) \( 9.84 \times 10^5 \text{ J mol}^{-1} \)

(b) \( 8.51 \times 10^5 \text{ J mol}^{-1} \)

(c) \( 6.56 \times 10^5 \text{ J mol}^{-1} \)

(d) \( 7.56 \times 10^5 \text{ J mol}^{-1} \)

Detailed Step-by-Step Solution

To solve this problem, we will use Bohr's model of the hydrogen atom. The ionization enthalpy is the energy required to remove an electron completely from the ground state (\(n=1\)) to infinity (\(n=\infty\)).

Step 1: Understand the Energy Formula

The energy of an electron in the \(n^{\text{th}}\) orbit of a hydrogen atom is given by:

\( E_n = -\frac{\text{Ionization Energy}}{n^2} \)

We are given the Ionization Energy (I.E.) = \( 1.312 \times 10^6 \text{ J mol}^{-1} \).

Step 2: Calculate Energy of the Initial and Final States

For the ground state (\(n=1\)):

\( E_1 = -\frac{1.312 \times 10^6}{1^2} = -1.312 \times 10^6 \text{ J mol}^{-1} \)

For the first excited state (\(n=2\)):

\( E_2 = -\frac{1.312 \times 10^6}{2^2} = -\frac{1.312 \times 10^6}{4} \)

\( E_2 = -0.328 \times 10^6 \text{ J mol}^{-1} \)

Step 3: Calculate the Excitation Energy (\(\Delta E\))

The energy required to excite the electron from \(n=1\) to \(n=2\) is the difference between their energies:

\( \Delta E = E_2 - E_1 \)

\( \Delta E = (-0.328 \times 10^6) - (-1.312 \times 10^6) \)

\( \Delta E = 1.312 \times 10^6 - 0.328 \times 10^6 \)

\( \Delta E = 0.984 \times 10^6 \text{ J mol}^{-1} \)

To match the options, we can rewrite this in standard scientific notation:

\( \Delta E = 9.84 \times 10^5 \text{ J mol}^{-1} \)

Conclusion: The energy required to excite the electron is \( 9.84 \times 10^5 \text{ J mol}^{-1} \). Therefore, the correct option is (a).

Deepen Your Knowledge of Atomic Structure

Understanding the energy levels of hydrogen-like species is a fundamental part of the Bohr Model. This concept frequently appears in both board exams and medical/engineering entrance tests. For a deep dive into atomic models, quantum numbers, and electron excitation, study our comprehensive guide on the Structure of Atom Class 11 Chemistry.

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