The ratio of the magnitude of the kinetic energy to the potential energy of an electron in the 5th$5^{\text{th}}$ excited state of a hydrogen atom is :

Related Formula

where n$n$ is the principal quantum number of the starting energy level.

Core Logic

For a hydrogen sample initially in the n = 4$n = 4$ level, the possible downward transition pathways to reach the ground state (n=1$n=1$) are:

Using the combination formula for all transitions:

Thus, 6 distinct spectral lines are generated.

Pattern Recognition

Think of it as counting combinations of transitions between levels: ₙC₂$_nC_2$. For n=4$n=4$, ₄C₂ = 6$_4C_2 = 6$ lines.

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Class 12 Physics: Atoms

Related Formula

The energy of the emitted photon during an atomic transition between energy states is given by:

where h$h$ is Planck's constant, c$c$ is speed of light, and λ$\lambda$ is the photon wavelength.

Core Logic

Write equations for the energy transitions from the layout of levels

Step 1: Finding Transition A to B

Subtracting equation (ii)$(ii)$ from equation (i)$(i)$ gives the net transition energy from A to B :

Pattern Recognition

Energy differences add linearly, which means their corresponding inverse wavelengths satisfy a parallel reciprocal subtraction rule: 1λAB = 1λAC - 1λBC$\frac{1}{\lambda_{AB}} = \frac{1}{\lambda_{AC}} - \frac{1}{\lambda_{BC}}$.

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Class 12 Physics: Atoms

Related Formula

The orbital frequency of revolution f$f$ of an electron is inversely proportional to its time period T$T$:

In Bohr's Atomic Model:

• Velocity v ∝ (Z)/(n)$v \propto \frac{Z}{n}$
• Radius r ∝ (n²)/(Z)$r \propto \frac{n^2}{Z}$

Core Logic

Substitute the proportional relationships of v$v$ and r$r$ into the frequency expression:

Step 1: Verification

Thus, the frequency varies inversely with the cube of the principal quantum number: f ∝ (1)/(n³)$f \propto \frac{1}{n^3}$.

Pattern Recognition

Remember the sequence of powers of n$n$ in Bohr's model: radius expands as n²$n^2$, velocity drops as n⁻¹$n^{-1}$, angular momentum grows as n¹$n^1$, and orbital time period or frequency changes as n³$n^3$ or n⁻³$n^{-3}$ respectively.

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Class 12 Physics: Atoms

Related Formula

Bohr state energy level values:

Transition excitation requirement:

Core Logic

To emit radiation in the Balmer series, the hydrogen electron must first be excited to at least the n=3$n=3$ shell. This allows it to jump down to n=2$n=2$ and produce the first spectral line of the Balmer series.

Energy required to transition from ground state (n=1$n=1$) to n=3$n=3$:

Step 1: Calculate Energy Gap

Pattern Recognition

Balmer emissions always return down to n=2$n=2$. Thus, the initial excitation starting from ground state n=1$n=1$ must reach at least n=3$n=3$ to create a valid Balmer transition.

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Class 12 Physics: Atoms

Core Logic

• Statement I: According to Rutherford's planetary model of the atom, almost the entire mass of an atom and all of its positive charge are concentrated in a centrally located small volume called the nucleus, with electrons revolving around it. Thus, Statement I is true.
• Statement II: Thomson's model of the atom (plum-pudding model) describes the atom as a spherical cloud of positive charge with electrons embedded in it. This model was proposed before Rutherford's nuclear model and is not a special case of Rutherford's model. Thus, Statement II is false.

Step 1: Conclusion

Therefore, Statement I is true but Statement II is false.

Pattern Recognition

Understand the evolutionary timeline of atomic models:

• Thomson's Plum Pudding Model (spherical cloud with embedded electrons)
• Rutherford's Nuclear Model (planetary orbital model with tiny centralized nucleus)

They are conceptually distinct, making Statement II false.

Chapter Mix

Class 12 Physics: Atoms