Problem 173 - A Complete Resource Book in Chemistry for JEE Main

Question

Calculate the wavelength (in monometer) associated with a proton moving at \(1.0 \times 10^{3} \mathrm{~ms}^{-1}\) (Mass of proton \(=1.67 \times 10^{-27} \mathrm{~kg}\) and \(\left.\mathrm{h}=6.63 \times 10^{-34} \mathrm{Js}\right): \quad[\mathbf]\) (a) \(0.40 \mathrm{~nm}\) (b) \(2.5 \mathrm{~nm}\) (c) \(14.0 \mathrm{~nm}\) (d) \(0.032 \mathrm{~nm}\)

Step-by-Step Solution

Answer

The wavelength is approximately 0.40 nm, corresponding to option (a).

Step 1: Identify the formula required

To find the wavelength of a particle, we use the de Broglie wavelength formula: \( \lambda = \frac{h}{mv} \), where \( h \) is Planck's constant, \( m \) is the mass of the particle, and \( v \) is its velocity.

Step 2: Substitute the given values

Using the formula for de Broglie wavelength, substitute the given values: \( h = 6.63 \times 10^{-34} \, \text{Js} \), \( m = 1.67 \times 10^{-27} \, \text{kg} \), and \( v = 1.0 \times 10^{3} \, \text{ms}^{-1} \). The equation becomes: \[ \lambda = \frac{6.63 \times 10^{-34}}{1.67 \times 10^{-27} \times 1.0 \times 10^{3}} \]

Step 3: Perform the calculation

Calculate the value of the denominator: \( m \times v = 1.67 \times 10^{-27} \times 1.0 \times 10^{3} = 1.67 \times 10^{-24} \)Now divide Planck's constant by this result to get the wavelength:\[ \lambda = \frac{6.63 \times 10^{-34}}{1.67 \times 10^{-24}} = 3.97 \times 10^{-10} \, \text{m} \]

Step 4: Convert units to nanometers

Since the problem asks for the wavelength in nanometers, convert from meters to nanometers (1 nanometer = \( 10^{-9} \) meters):\[ 3.97 \times 10^{-10} \, \text{m} = 0.397 \, \text{nm} \]

Step 5: Choose the correct answer

Compare the calculated wavelength, \(0.397 \, \text{nm}\), to the options provided:(a) \(0.40 \, \text{nm}\)(b) \(2.5 \, \text{nm}\)(c) \(14.0 \, \text{nm}\)(d) \(0.032 \, \text{nm}\)The closest answer is (a) \(0.40 \, \text{nm}\).

Key Concepts

Planck's Constant

Planck's constant, often symbolized by \( h \), is a fundamental quantity in quantum physics. It plays a crucial role in the theories concerning the behavior of matter at microscopic scales. Planck's constant has the value \( 6.63 \times 10^{-34} \mathrm{~Js} \) and is a cornerstone in the equation for the de Broglie wavelength:

• It relates the energy of a photon to the frequency of its electromagnetic wave using the formula \( E = h u \), where \( u \) is the frequency.
• In the context of wave-particle duality, it allows one to calculate the wavelength of a particle, linking wave characteristics with particle mechanics.

This tiny constant signifies that quantum effects are generally noticeable only at atomic and subatomic scales. However, it is indispensable for calculations that involve particles acting like waves.

Proton Mass

The proton mass is a critical component in calculating the de Broglie wavelength of particles like protons and electrons. It's approximately \( 1.67 \times 10^{-27} \mathrm{~kg} \). Understanding proton mass helps in:

• Determining how massive particles, compared to lighter ones like electrons, have significantly shorter wavelengths at the same velocity, thus experiencing reduced wave-like behaviors.
• Providing necessary context for how the mass impacts wavelength, given that a larger mass requires less wavelength for a constant velocity, as shown in the equation \( \lambda = \frac{h}{mv} \).

In our specific calculation, the proton's mass allows us to determine its wave properties as it moves, emphasizing how mass and motion impact de Broglie wavelengths.

Velocity of a Particle

The velocity of a particle is a fundamental parameter in determining its de Broglie wavelength. Defined in meters per second \( \mathrm{m/s} \), velocity is a measure of how fast and in which direction a particle is moving. It greatly influences the computation of wavelength because:

• The velocity of the particle, alongside its mass, directly affects the wavelength. With higher velocities, particles like protons have shorter wavelengths.
• This relationship underscores the concept of wave-particle duality, where faster-moving particles exhibit less noticeable wave characteristics.
• In our problem, a velocity of \( 1.0 \times 10^{3} \mathrm{~ms}^{-1} \) was provided to calculate the proton's wavelength accurately using de Broglie's formulation \( \lambda = \frac{h}{mv} \).

The velocity's impact on wavelength highlights the intertwined nature of mass, speed, and wave behavior, showcasing how motion affects the quantum characteristics of particles.