Ionisation energy of He^+ is 19.6 ×10^-18 J atom ^-1. The energy of the first stationary state

step 1 The question is ionization energy of helium plus is the energy of first stationary state of LI2

Ionisation energy of He^+ is 19.6 ×10^-18 J atom ^-1. The...

Ionisation energy of $\mathrm{He}^{+}$ is $19.6 \times 10^{-18} \mathrm{~J}$ atom $^{-1}$. The energy of the first stationary state $(n=1)$ of $\mathrm{Li}^{2+}$ is : [A.I.E.E.E. 2010] (a) $8.82 \times 10^{-17} \mathrm{~J}$ atom $^{-1} \square$ (b) $4.41 \times 10^{-16} \mathrm{~J}$ atom $^{-1}$ (c) $-4.41 \times 10^{-17} \mathrm{~J}$ atom $^{-1} \square$ (d) $-2.2 \times 10^{-15} \mathrm{~J}$ atom $^{-1}$ [Hint : $\mathrm{E}_{\mathrm{H}}$ (for $\left.\mathrm{H}\right) \times \mathrm{Z}^{2}=\mathrm{I}$.E. or. $$ \begin{aligned} \mathrm{E}_{\mathrm{H}} \times 4 &=\text { l.E. of } \mathrm{He}^{+}=-19.6 \times 10^{-18} \mathrm{~J} \\ \mathrm{E}_{\mathrm{H}} &=-\frac{19.6 \times 10^{-18}}{4} \mathrm{~J} \end{aligned} $$ $$ \begin{aligned} \mathrm{E}_{\mathrm{Li}^{2+}} &=\mathrm{E}_{\mathrm{H}} \times 9=-\frac{19.6 \times 10^{-18}}{4} \times 9 \\ &\left.=-44.1 \times 10^{-18} \mathrm{~J}=-4.41 \times 10^{-17} \mathrm{~J}\right] \end{aligned} $$

Ionisation energy of $\mathrm{He}^{+}$ is $19.6 \times 10^{-18} \mathrm{~J}$ atom $^{-1}$. The energy of the first stationary state $(n=1)$ of $\mathrm{Li}^{2+}$ is :
[A.I.E.E.E. 2010]
(a) $8.82 \times 10^{-17} \mathrm{~J}$ atom $^{-1} \square$
(b) $4.41 \times 10^{-16} \mathrm{~J}$ atom $^{-1}$
(c) $-4.41 \times 10^{-17} \mathrm{~J}$ atom $^{-1} \square$
(d) $-2.2 \times 10^{-15} \mathrm{~J}$ atom $^{-1}$
[Hint : $\mathrm{E}_{\mathrm{H}}$ (for $\left.\mathrm{H}\right) \times \mathrm{Z}^{2}=\mathrm{I}$.E.
or.
$$
\begin{aligned}
\mathrm{E}_{\mathrm{H}} \times 4 &=\text { l.E. of } \mathrm{He}^{+}=-19.6 \times 10^{-18} \mathrm{~J} \\
\mathrm{E}_{\mathrm{H}} &=-\frac{19.6 \times 10^{-18}}{4} \mathrm{~J}
\end{aligned}
$$
$$
\begin{aligned}
\mathrm{E}_{\mathrm{Li}^{2+}} &=\mathrm{E}_{\mathrm{H}} \times 9=-\frac{19.6 \times 10^{-18}}{4} \times 9 \\
&\left.=-44.1 \times 10^{-18} \mathrm{~J}=-4.41 \times 10^{-17} \mathrm{~J}\right]
\end{aligned}
$$

Key Concepts

Nuclear Charge (Z) Scaling

The energy levels in hydrogen-like atoms depend on the square of the nuclear charge (Z^2). This means that an increase in the number of protons in the nucleus leads to a stronger coulombic attraction, thereby making the electron more tightly bound and increasing the energy required to remove it. This Z^2 scaling is a critical factor in determining the spectral lines and ionisation energies of these ions.

Ionisation Energy

Ionisation energy is the minimum energy required to remove an electron from an atom or ion, effectively moving it from a bound state to a state of zero binding energy. For hydrogen-like ions, this is equivalent to the absolute value of the energy of the electron in the lowest energy level (n=1), reflecting the strength of the attraction between the electron and the nucleus.

Quantized Energy Levels

In quantum mechanics, electrons in atoms can only occupy specific, discrete energy levels characterized by a principal quantum number, n. For hydrogen-like atoms, the energy of each level is determined by the formula E_n = -E_0*(Z^2/n^2), where E_0 is the ground state energy of hydrogen, Z is the atomic number, and n is the principal quantum number. This quantization explains why atoms emit or absorb energy in fixed amounts.

Hydrogen-like Atoms

Hydrogen-like atoms are systems that contain only one electron, regardless of the number of protons in the nucleus. Because of this, their energy levels can be calculated using models similar to that of the hydrogen atom, with appropriate adjustments for the nuclear charge. This makes the energy level structure inherently similar across different species, allowing for formulas based on the nuclear charge to be used.

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