Problem 130 - A Complete Resource Book in Chemistry for JEE Main

Question

The wavelength associated with an electron (mass = \(9.11 \times 10^{-31} \mathrm{~kg}\) ) moving with a velocity of \(10^{6} \mathrm{~m} \mathrm{~s}^{-1}\) is \(\left(\mathrm{h}=6.625 \times 10^{-34} \mathrm{~J} \mathrm{~s}\right)\) (a) \(0.727 \mathrm{~nm}\) (b) \(7.27 \mathrm{~nm}\) (c) \(727 \mathrm{~nm}\) (d) \(7.27 \mathrm{~m}\)

Step-by-Step Solution

Answer

The wavelength is 0.727 nm, matching option (a).

Step 1: Identify the Formula

To find the wavelength associated with an electron, we use the de Broglie wavelength formula: \[\lambda = \frac{h}{mv}\]where \( \lambda \) is the wavelength, \( h \) is the Planck's constant \((6.625 \times 10^{-34} \mathrm{~J} \mathrm{~s})\), \( m \) is the mass of the electron \((9.11 \times 10^{-31} \mathrm{~kg})\), and \( v \) is the velocity \((10^{6} \mathrm{~m/s})\).

Step 2: Substitute Known Values

Substitute the given values into the de Broglie wavelength formula:\[\lambda = \frac{6.625 \times 10^{-34}}{9.11 \times 10^{-31} \times 10^{6}}\]

Step 3: Calculate the Wavelength

Perform the calculation:1. Calculate the denominator: \[9.11 \times 10^{-31} \times 10^{6} = 9.11 \times 10^{-25}\]2. Divide Planck's constant by the result from the previous step: \[\lambda = \frac{6.625 \times 10^{-34}}{9.11 \times 10^{-25}} = 7.27 \times 10^{-10} \mathrm{~m}\]

Step 4: Convert Meters to Nanometers

Since the options are given in nanometers (nm), convert the result from meters to nanometers:1 nm = \(10^{-9} m\).\[ \lambda = 7.27 \times 10^{-10} \mathrm{~m} \times \frac{1 \mathrm{~nm}}{10^{-9} \mathrm{~m}} = 0.727 \mathrm{~nm} \]

Step 5: Compare with Options

Compare the calculated wavelength with the multiple choice options. The result of \(0.727 \mathrm{~nm}\) matches option (a).

Key Concepts

Planck's constant

Planck's constant is a fundamental constant in quantum mechanics, symbolized as \( h \). It has a value of \(6.625 \times 10^{-34} \, \mathrm{J \cdot s} \). This constant is crucial in the formulation of the de Broglie wavelength, which relates to the wave-like properties of particles. Originally introduced by Max Planck, this constant is used to describe how quantum objects like electrons can have both particle and wave characteristics.

• Planck's constant helps us bridge classical and quantum physics.
• It plays a vital role in determining the energy of photons: \( E = h u \), where \( u \) is the frequency.
• In the de Broglie equation, it's used to find wavelengths associated with moving particles.

Recognizing the small magnitude of Planck's constant underscores why wave-like phenomena are not normally observed at the macroscopic scale and primarily seen in tiny particles like electrons.

Electron Mass

The mass of an electron is a fundamental property critical for calculations in physics and chemistry. The electron mass is approximately \(9.11 \times 10^{-31} \, \mathrm{kg} \). This tiny mass is why electrons exhibit pronounced wave-like behavior. Because their mass is so small, their de Broglie wavelengths are large enough to be relevant in microscopic systems.

• Electron mass is a key parameter in determining the wavelength in quantum mechanical systems.
• The tiny mass allows electrons to behave according to quantum mechanics rather than classical physics.
• In the de Broglie equation, mass directly affects the calculated wavelength, with the formula \( \lambda = \frac{h}{mv} \).

Understanding electron mass helps explain the vast difference in behavior between electrons and much heavier particles in everyday life.

Velocity of Electron

The velocity at which an electron travels is a central factor in calculating its associated de Broglie wavelength. In the given exercise, the electron velocity is \(10^{6} \, \mathrm{m/s} \). This speed is within a typical range for electrons, allowing them to exhibit significant wave properties and making concepts like de Broglie wavelength applicable.

• The velocity of an electron is used alongside its mass and Planck's constant in the de Broglie formula: \( \lambda = \frac{h}{mv} \).
• Higher velocities correspond to shorter wavelengths, demonstrating how velocity impacts wave characteristics.
• Electron speed in materials affects their electrical and thermal conductivity properties.

The velocity is vital as it influences the behavior of electrons in different materials and conditions, which is crucial in fields like electronics and quantum mechanics.