Question

The number of waves in 3 rd orbit of \(\mathrm{H}\) -atom is: (a) 1 (b) 2 (c) 3 (d) 4

Short answer

The number of waves in the 3rd orbit of the H-atom is 3.

Step-by-step solution

Understanding the Concept of Waves in Orbits

The wave number in a stationary orbit correlates with the quantum number of the electron in that orbit. For any hydrogen atom, the number of waves in an orbit is equal to its principal quantum number (n).

Identify the Principal Quantum Number

For the hydrogen atom in question, we are tasked with finding the number of waves in the 3rd orbit. The principal quantum number for the 3rd orbit is 3.

Conclusion on the Number of Waves

Since the principal quantum number is 3, the number of complete de Broglie wavelength waves in the 3rd orbit is also 3.

Key concepts

Quantum Number

Quantum numbers are like the addresses of electrons in an atom. They help in locating and describing the specific energy states of electrons around the nucleus. In the hydrogen atom, the quantum number, more specifically the principal quantum number, tells us about the energy level or shell an electron occupies.
The principal quantum number is denoted by the symbol "n". It can take any positive integer value like 1, 2, 3, etc. This number indicates the distance of the electron from the nucleus.

• The higher the value of "n", the farther the electron is from the nucleus, and the higher the energy level.
• For instance, when "n" is 1, it describes the electron at the closest energy level to the nucleus, called the first orbit or shell.
• In this case, for a hydrogen atom's third orbit, the principal quantum number is 3. This means the electron is in the third energy level, and the orbit has three waves or cycles of the electron wave. It’s a bit like saying there are three complete Ferris wheels, each representing a full cycle of the electron's energy!

de Broglie Wavelength

The de Broglie wavelength is a fascinating concept that merges the world of particles with waves. Proposed by Louis de Broglie, it suggests that matter, like electrons, can exhibit wavelike properties. This concept is essential in understanding the behavior of electrons in atoms.
The formula for the de Broglie wavelength \( \lambda \) is given by \[ \lambda = \frac{h}{mv} \] where \( h \) is Planck's constant, \( m \) is the mass of the particle, and \( v \) is its velocity.

• It shows that an electron is not just a particle but also has a wavelength associated with it. Talking about the de Broglie wavelength of the hydrogen atom in the 3rd orbit naturally brings the quantum numbers into context, as they determine the number of complete wavelengths around the orbit.
• In the third orbit of a hydrogen atom, there are precisely three de Broglie wavelengths fitting around the path.
• This wave-particle duality makes it possible to understand how electrons can exist within specified quantized (distinct) orbits or shells, where their wavelengths fit perfectly, like a string looping just the right amount of times into a circle.

Wave Concept in Bohr Model

The Bohr model of the atom introduced a revolutionary way of looking at the atom, combining classical and quantum ideas. It suggested that electrons revolve in fixed orbits around the nucleus, much like planets around the sun, but with some fascinating quantum rules intertwined.
The Bohr model showed that these orbits are quantized, meaning electrons can only occupy certain allowed orbits where their wave nature fits perfectly. This understanding was deepened with the wave concept.

• The orbit an electron takes corresponds to a whole number of de Broglie wavelengths fitting precisely into the orbit, which aligns perfectly with their quantum number, "n".
• For the third orbit, this means there are three full de Broglie wavelengths present, creating stable, permissible paths for electrons.
• This concept helps explain why electrons do not spiral into the nucleus but instead stay in stable orbits - they are in a "wave state" where the conditions match perfectly, allowing electrons to "reside" in these orbits without falling off or gaining unwanted energy levels unexpectedly.
• The wave concept in the Bohr model paints a picture of the atom as a harmonious system where waves and particles coexist without contradiction.