A \(1.0 \mathrm{~g}\) particle is shot from a gun with velocity of \(100 \mathrm{~m} / \mathrm{sec}\). Calculate its de Broglie wavelength.

Planck's constant, symbolized by the letter 'h', is a fundamental quantity in quantum mechanics. Its value is approximately equal to \(6.62607015 \times 10^{-34} \mathrm{Js}\). This constant is crucial because it relates the energy of a photon to the frequency of the light. Mathematically, this relationship is expressed as \(E = h u\), where 'E' represents energy and '\(u\)' is the frequency. In the context of de Broglie wavelength calculations, Planck's constant is pivotal since it allows us to establish a link between the wave-like and particle-like properties of matter.

Understanding the importance of Planck's constant for quantum physical phenomena, therefore, becomes essential for solving problems that bridge classical and quantum views, such as those found in IIT-JEE Physical Chemistry.

Momentum in physics is described as the product of a particle's mass and velocity and is a key concept in both classical and quantum mechanics. If we denote mass by 'm' and velocity by 'v', the equation for momentum 'p' is written as \(p = m \times v\). Momentum is a vector quantity, which means that it has both magnitude and direction, decisive elements in understanding the motion of particles.

In de Broglie's equation, the momentum plays a central role as it connects the particle's mechanical properties with its wavelike characteristics. When attempting to conceptualize a particle's movement in quantum scales, appreciating the influence of momentum is essential for grasping the underlying physics of the phenomenon.

The concept of a particle's wavelength, also known as the de Broglie wavelength, bridges the gap between classical physics and quantum mechanics. It introduces the wave-particle duality of matter, that all particles exhibit both wave and particle characteristics. The de Broglie wavelength is given by \(\lambda = \frac{h}{p}\), where '\(\lambda\)' is the wavelength, 'h' is Planck's constant, and 'p' is the momentum.

For students tackling physical chemistry problems, such as those in IIT-JEE, understanding how to calculate a particle's wavelength is crucial because it provides insights into the behavior of electrons, protons, and other subatomic particles. Factors like particle speed and mass directly influence the wavelength, and hence the quantum behavior of particles can be inferred from these classical properties.

Students preparing for competitive exams like the Indian Institutes of Technology - Joint Entrance Examination (IIT-JEE) often encounter problems involving physical chemistry that require a deep understanding of both theoretical concepts and their practical application. Questions involving de Broglie wavelength calculations are common in physical chemistry sections because they incorporate concepts of quantum mechanics in explaining the behavior of atoms and molecules.

The ability to solve such problems depends on understanding formulas and constants such as Planck's constant and relating them to tangible physical properties like momentum. Mastery of these related concepts allows students to make connections between seemingly disparate ideas, a skill that is tested intensively in the Physical Chemistry problems of IIT-JEE.