An electron travels with a velocity of \(x \mathrm{~ms}^{-1}\). For a proton to have the same de-Broglie wavelength, the velocity will be approximately: (a) \(\frac{1840}{x}\) (b) \(\frac{x}{1840}\) (c) \(1840 x\) (d) \(x\)

One of the fundamental principles of quantum mechanics is the de-Broglie hypothesis, which introduces the concept that matter can exhibit both particle and wave-like properties. This duality is quantitatively expressed by the de-Broglie wavelength equation:

\[ \text{De-Broglie Wavelength, } \boldsymbol{\text{\textlambda}} = \frac{h}{mv} \]
where \( h \) is known as Planck's constant, \( m \) refers to the mass of the particle, and \( v \) is its velocity. The greater the mass or velocity of a particle, the shorter its de-Broglie wavelength, manifesting more particle-like behavior, while lighter or slower particles exhibit longer wavelengths, indicating wavelike characteristics.

For example, when applying this equation to an electron, which has a relatively small mass, even modest velocities result in a quantifiable wavelength, allowing us to observe its wave-like nature. The ability to calculate the de-Broglie wavelength is essential in areas such as electron microscopy and quantum field theory.

The masses of subatomic particles are a central consideration in many physics and chemistry calculations, as in the de-Broglie wavelength problem. Electrons and protons are two of the most studied particles, with distinct differences in mass. An electron has a relatively small mass (approximately \(9.109 \times 10^{-31}\text{kg})\text{,}\) and a proton is much more massive (about \(1.673 \times 10^{-27}\text{kg})\text{,}\) which is roughly 1840 times heavier than the electron.

This significant difference in mass affects how they behave under similar conditions. For instance, since the proton is so much heavier, we would expect it to move more slowly than an electron if both had to display the same de-Broglie wavelength. The variation in mass is also why protons contribute more to the atomic mass than electrons, a concept that is crucial in chemistry, particularly in balancing reactions and understanding molecular structures.

Planck's constant, denoted as \( h \text{,}\) is one of the key elements in quantum mechanics. Its value is \(6.626 \times 10^{-34}\text{Js}\text{ (joule-seconds}).\) This constant is what connects the energy of a photon (the quantum of electromagnetic radiation) to its frequency through the equation \(E = hf\text{,}\) where \(E\) is energy and \(f\) is frequency.

Planck's constant is also pivotal in the de-Broglie equation as it relates a moving particle's momentum to its wavelength. The unvarying nature of this constant allows for precise calculations and is foundational in studies that range from the smallest scales in quantum physics to the expansive reaches of cosmology. It forms a fundamental part of understanding how the universe behaves on a subatomic level.

The Joint Entrance Examination (JEE) is a highly competitive test in India for students aspiring to enter engineering colleges. Physical Chemistry is a critical subject within the JEE Chemistry section. It involves the study of subatomic particles, thermodynamics, chemical kinetics, equilibrium, and quantum chemistry, including concepts like the de-Broglie wavelength.

Understanding the de-Broglie wavelength equation and its applications can be instrumental in solving JEE Physical Chemistry problems, as it integrates the principles of both physics and chemistry. Mastery of this topic not only helps in cracking competitive exams like JEE but also lays a solid foundation for future scientists and engineers who may want to delve deeper into materials science, nanotechnology, and various other fields of research.