Consider the spectral line resulting from the transition n =2 to n=1 in the atoms and ions given below. The shortest wavelength is produced by:A. Hydrogen atomB. Deuterium atomC. Singly ionised heliumD. Doubly ionised lithium

The correct answer is D. Doubly ionised lithium.
Explanation:
The wavelength of a spectral line in hydrogen-like atoms or ions can be calculated using the Rydberg formula:
1λ=RZ2(1n12−1n22)\frac{1}{\lambda} = R Z^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)
Where:
• λ\lambda: Wavelength of the spectral line.
• RR: Rydberg constant (1.097×107 m−11.097 \times 10^7 \, \text{m}^{-1}).
• ZZ: Atomic number of the atom or ion.
• n1n_1: Lower energy level (here, n1=1n_1 = 1).
• n2n_2: Higher energy level (here, n2=2n_2 = 2).
Key Points:
• The wavelength of the emitted photon is inversely proportional to Z2Z^2.
• A higher ZZ (atomic number) results in a stronger nuclear attraction, leading to greater energy level separation, which produces photons with shorter wavelengths during transitions.
Analysis of Each Option:
1. Hydrogen atom (Z=1Z = 1): For hydrogen, the emitted photon wavelength is based on Z=1Z = 1. It has the longest wavelength compared to the other options because Z2=12=1Z^2 = 1^2 = 1.
2. Deuterium atom (Z=1Z = 1): Deuterium is an isotope of hydrogen, with Z=1Z = 1 (same atomic number as hydrogen). The wavelength is almost identical to that of hydrogen since the nuclear charge remains the same, and the isotopic mass difference does not significantly affect the spectral transition.
3. Singly ionised helium (Z=2Z = 2): For singly ionised helium (He+\text{He}^+), the nucleus has Z=2Z = 2. The energy level separations are proportional to Z2=22=4Z^2 = 2^2 = 4, leading to shorter wavelengths than hydrogen or deuterium.
4. Doubly ionised lithium (Z=3Z = 3): For doubly ionised lithium (Li2+\text{Li}^{2+}), the nucleus has Z=3Z = 3. The energy level separations are proportional to Z2=32=9Z^2 = 3^2 = 9, resulting in the shortest wavelength among the given options.
Conclusion:
The shortest wavelength is produced by doubly ionised lithium (Li2+\text{Li}^{2+}), as its atomic number Z=3Z = 3 creates the largest energy level separation.
Thus, the correct answer is:
D. Doubly ionised lithium\boxed{\text{D. Doubly ionised lithium}}