Problem 157 - A Complete Resource Book in Chemistry for JEE Main

Question

The de Broglie wavelength of a tennins ball of mass \(60 \mathrm{~g}\) moving with a velocity of 10 metres per second is approximately \([2003]\) Planck constant \(\mathrm{h}=6.63 \times 10^{-31} \mathrm{Js}\) ) (a) \(10^{-33} \mathrm{~m}\) (b) \(10^{-31} \mathrm{~m}\) (c) \(10-16 \mathrm{~m}\) (d) \(10^{-25} \mathrm{~m}\)

Step-by-Step Solution

Answer

The de Broglie wavelength is approximately \( 10^{-31} \text{ m} \) (option b).

Step 1: Identify the formula

The de Broglie wavelength \( \lambda \) is given by the formula \( \lambda = \frac{h}{mv} \), where \( h \) is the Planck constant, \( m \) is the mass of the object, and \( v \) is its velocity.

Step 2: Convert mass to kilograms

Since the mass is given in grams, convert it to kilograms: \( 60 \text{ g} = 0.06 \text{ kg} \).

Step 3: Substitute values into the formula

Use the values provided: \( h = 6.63 \times 10^{-31} \text{ Js} \), \( m = 0.06 \text{ kg} \), and \( v = 10 \text{ m/s} \). Substitute these into the formula: \( \lambda = \frac{6.63 \times 10^{-31} \text{ Js}}{0.06 \text{ kg} \times 10 \text{ m/s}} \).

Step 4: Calculate the denominator

Calculate the product in the denominator: \( 0.06 \text{ kg} \times 10 \text{ m/s} = 0.6 \text{ kg m/s} \).

Step 5: Calculate the wavelength

Divide the Planck constant by the result from Step 4: \( \lambda = \frac{6.63 \times 10^{-31} \text{ Js}}{0.6 \text{ kg m/s}} = 1.105 \times 10^{-31} \text{ m} \).

Key Concepts

Planck constant

The Planck constant, denoted as \( h \), is a fundamental constant in physics that plays a crucial role in quantum mechanics. It is equivalent to \( 6.63 \times 10^{-34} \) Joule seconds (Js). This small value demonstrates how the quantum realm operates on a scale vastly different from our everyday experiences. The Planck constant characterizes the size of the smallest action or quantum of action that can be observed.
In the context of de Broglie's hypothesis, the Planck constant is used to relate a particle's momentum to its wavelength. The formula \( \lambda = \frac{h}{mv} \) shows how \( h \) is essential in calculating the wavelength associated with an object in motion. If we take, for instance, a tiny particle like an electron, its wavelength will be significant enough to consider. However, for larger objects like a tennis ball, the de Broglie wavelength becomes nearly imperceptible due to the larger mass involved.

Mass Conversion

When we discuss the de Broglie wavelength, we need to ensure that all units are consistent. This is true especially for mass, which might be initially given in grams. In physics calculations, we often convert mass into kilograms to maintain compatibility with the International System of Units (SI units), ensuring calculations are correct.
In our example, the mass of the tennis ball is given as 60 grams. This mass needs to be converted to kilograms, as the Planck constant and velocity are in units of Joules and meters respectively. The conversion is straightforward: 1 gram equals 0.001 kilograms, so 60 grams equals 0.06 kilograms. This figure is then utilized to find the de Broglie wavelength of the tennis ball.

Quantum Mechanics

Quantum mechanics is a branch of physics dealing with phenomena on very small scales, such as atoms and subatomic particles. It describes the peculiar and sometimes non-intuitive ways that particles and energy interact at these scales. Here, traditional physics is replaced by probabilities and wave-like behaviors.
In the framework of quantum mechanics, Louis de Broglie proposed that particles can also exhibit wave-like properties. This duality is where de Broglie's hypothesis comes in. It's quite astonishing that an object we typically see as solid, like a tennis ball, can have a wavelength, albeit incredibly tiny. This is why quantum mechanics often refers to particles like electrons, where such wave characteristics become influential and readily observable.
As the size of an object increases, its wavelength decreases due to the multiplication of mass and velocity—illustrating why quantum effects are not noticeable on the human scale. Thus, quantum mechanics brings forward the spectral dance of particles in ways unseen, until translated through mechanisms like the de Broglie wavelength.