x different wavelengths may be observed in the spectrum from a hydrogen sample if the atoms are exit

Q: Why do we use the formula $n(n-1)/2$?

Because each spectral line corresponds to a transition between two distinct energy levels. Choosing 2 levels out of $n$ available levels is given by the combination formula $^nC_2$, which simplifies to $n(n-1)/2$.

Q: What if the question asked for a single hydrogen atom instead of a sample?

A single atom has only one electron. It can only take one specific path down to the ground state. The maximum number of different wavelengths it could emit is $n-1$ (e.g., $6 \to 5 \to 4 \to 3 \to 2 \to 1$).

Q: Are all these wavelengths in the visible region?

No, transitions ending at $n=1$ are in the ultraviolet (Lyman series), $n=2$ are visible (Balmer series), and $n=3, 4, 5$ are in the infrared region.

Q: Does the number of lines depend on the element?

The formula for the *number* of lines depends only on the principal quantum number $n$. However, the actual *values* of the wavelengths will depend on the atomic number $Z$ of the element.

Q: Can an electron jump directly from $n=6$ to $n=1$?

Yes, that is one of the possible transitions, and it corresponds to the highest energy (and shortest wavelength) photon emitted in this scenario.

Question

$x$ different wavelengths may be observed in the spectrum from a hydrogen sample if the atoms are exited to states with principal quantum number $n = 6$? The value of $x$ is ........ .

Sigma Percentile · Accepted Answer

The phenomenon of atomic emission spectra is one of the most beautiful validations of the Bohr model of the atom. When a sample of hydrogen gas is excited—perhaps by passing an electric discharge through it or heating it—the electrons in the atoms absorb energy and jump to higher, unstable energy levels.

But what goes up must come down. As these electrons return to lower energy states, they release the excess energy in the form of photons. Each specific jump corresponds to a photon of a specific energy, and therefore, a specific wavelength.

### The Combinatorics of Spectral Lines

Imagine an electron sitting at the $n=6$ energy level. It doesn't have to jump all the way down to $n=1$ in a single leap. It could jump to $n=5$, then to $n=2$, and finally to $n=1$. Each of these intermediate jumps emits a distinct wavelength of light.

Because we are dealing with a **sample** of hydrogen gas, we have billions of atoms. Statistically, every possible transition between the available energy levels will occur in some atom within the sample.

To find the total number of unique spectral lines, we simply need to find the number of ways to choose a starting level and an ending level from the $n$ available levels. This is a classic combinations problem: choosing 2 items from $n$ items, denoted as $^nC_2$.

The formula for this is:
$$N = \frac{n(n-1)}{2}$$

### Calculating the Wavelengths

In our specific problem, the atoms are excited to the principal quantum number $n=6$. We substitute this into our master equation:

$$N = \frac{6(6-1)}{2}$$
$$N = \frac{6 	imes 5}{2}$$
$$N = \frac{30}{2} = 15$$

Thus, the sample will emit **15 distinct wavelengths** of light.

### The "Single Atom" Trap

A classic trap in JEE and NEET exams is to change one word in the problem: asking for the maximum number of lines from a **single isolated hydrogen atom** instead of a sample.

A single atom only has one electron. That single electron can only take one specific path down to the ground state. The path that produces the maximum number of distinct lines is a cascade through every single intermediate level (e.g., $6 	o 5 	o 4 	o 3 	o 2 	o 1$).

In this cascade, the number of jumps is simply $n-1$. So, for a single atom excited to $n=6$, the maximum number of lines would be $6-1 = 5$. Always read the question carefully to see if you are dealing with a bulk sample or a single atom!

Animated Solution for Physics - Atoms and Nuclei: $x$ different wavelengths may be observed in the spectrum from a hydrogen sample if the atoms are exited to states with principal quantum number $n = 6$ ? The value of $x$ is ........ .

Enter Numerical Value:

Visualized Solution

Energy Levels

Number of Spectral Lines

Substitution

Calculation

Final Answer

Single Atom vs Sample

The Sigma Insight: Bohr's Atomic Model and Energy Levels

The Combinatorics of Spectral Lines

Calculating the Wavelengths

The "Single Atom" Trap

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