Problem 131 - 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, option (a).

Step 1: Understanding De Broglie Wavelength Formula

The De Broglie wavelength \((\lambda)\) is given by the formula \(\lambda = \frac{h}{mv}\), where \(h\) is Planck's constant, \(m\) is the mass of the particle, and \(v\) is the velocity of the particle. We need to substitute the given values into this formula to find the wavelength.

Step 2: Substitute Given Values

Substitute the provided values into the formula: \(\lambda = \frac{6.625 \times 10^{-34} \text{ J s}}{9.11 \times 10^{-31} \text{ kg} \times 10^{6} \text{ m/s}}\).

Step 3: Calculate the Denominator

Calculate the denominator: \(9.11 \times 10^{-31} \times 10^{6} = 9.11 \times 10^{-25}\).

Step 4: Calculate the Wavelength

Now, divide Planck's constant by the result of the denominator: \(\lambda = \frac{6.625 \times 10^{-34}}{9.11 \times 10^{-25}}\).

Step 5: Final Calculation

Perform the division to find \(\lambda\): \(\lambda \approx \frac{6.625}{9.11} \times 10^{-34 + 25} = 0.727 \times 10^{-9}\) meters, which is \(0.727 \text{ nm}\).

Step 6: Verify Answer with Options

Check the result against the provided options. The calculated answer \(0.727 \text{ nm}\) matches with option (a), confirming it.

Key Concepts

Planck's constant

Planck's constant, denoted by the symbol \( h \), is a fundamental quantity in quantum mechanics. It bridges the macroscopic world with the quantum world by linking the energy of a photon to its frequency. The value of Planck's constant is \( 6.625 \times 10^{-34} \text{ J s} \). This tiny number plays a huge role in various quantum phenomena.

In the context of the de Broglie wavelength, Planck's constant shows up in calculating the wavelength associated with any moving particle. This discovery was pivotal in the early 20th century and challenged classical physics at that time. The constant essentially indicates how matter demonstrates both wave-like and particle-like properties.

• Planck's constant is a bridge between classical and quantum physics.
• It shows that energy is quantized, meaning it comes in discrete packets.

Its presence in the formula \( \lambda = \frac{h}{mv} \) underscores how every particle, no matter how small, carries a wave characteristic that can be calculated.

electron mass

The electron mass is one of the most crucial constants to know in quantum mechanics and even classical physics. It's approximately \( 9.11 \times 10^{-31} \text{ kg} \), which is incredibly tiny. This small mass means that electrons, due to their wave properties, can easily exhibit phenomena like diffraction and interference when moving through a medium.

In our everyday world, mass is typically associated with something tangible, but in the quantum realm, the electron's small mass lets it behave very differently from larger objects. In the de Broglie wavelength context, the mass of the electron is the denominator in the equation \( \lambda = \frac{h}{mv} \). This helps calculate the electron's wavelength, showcasing its dual nature as both wave and particle. The small mass significantly impacts the calculated wavelength, resulting in observable wave-like properties.

• Electron mass is fundamental to calculating the behavior of electrons.
• The tiny mass means electrons readily show wave-like properties.

velocity of particle

Velocity is a key component in determining the de Broglie wavelength of a particle. In physics, velocity refers to the speed of something in a given direction. For electrons, which are of such light mass, even a small increase in their velocity can significantly change their behavior.

In our exercise, the velocity of the electron was given as \( \text{10}^{6} \text{ m/s} \). This fast pace for such a light particle means its wavelength will be pronounced, making it easier to observe its wave-like behavior in experiments. This detail is critical when predicting how the electron will behave in a given situation, particularly in quantum experiments.

• Velocity determines how wave-like an electron will behave.
• Increased velocity results in shorter wavelengths due to the inverse relation in the formula \( \lambda = \frac{h}{mv} \).

By understanding velocity, we get a clearer picture of the electron's movement and its wave characteristics.

quantum mechanics

Quantum mechanics is the branch of physics that deals with phenomena at the smallest scales, such as atoms and subatomic particles like electrons. It introduces the idea that particles can exist in different states simultaneously and that properties like position and momentum are not always certain but rather probabilistic.

De Broglie's groundbreaking idea was that particles like electrons have wavelengths and could thus exhibit wave properties such as interference and diffraction. This was a major shift from classical physics which viewed matter only as particles.

• Quantum mechanics explains the dual nature of matter, accommodating both wave and particle characteristics.
• This branch of physics also introduced the concept of quantization, where certain properties can only take on discrete values.
• Understanding quantum mechanics helps make sense of why electrons exhibit wave-like properties and enables the accurate prediction of phenomena at the atomic scale.

In solving the de Broglie wavelength problem, you are using principles of quantum mechanics to predict particle behavior, foundational for many modern technologies.