Question

Let \( \lambda_e, \lambda_p, \lambda_d \) be the wavelengths associated with an electron, a proton, and a deuteron, all moving with the same speed. Then the correct relation between them is:

• A. \( \lambda_e>\lambda_p>\lambda_d \)
• B. \( \lambda_p>\lambda_e>\lambda_d \)
• C. \( \lambda_e>\lambda_p>\lambda_d \)
• D. \( \lambda_e = \lambda_p = \lambda_d \)

Hint:

The de Broglie wavelength is inversely proportional to the mass of the particle. Heavier particles have smaller wavelengths.

Answer 1

The Correct Option is A

To establish the relationship between the de Broglie wavelengths (\( \lambda_e, \lambda_p, \lambda_d \)) for an electron, proton, and deuteron, all traveling at the same velocity, we employ the de Broglie wavelength formula: \( \lambda = \frac{h}{mv} \). In this equation, \( \lambda \) represents the wavelength, \( h \) is Planck's constant, \( m \) is the particle's mass, and \( v \) is its velocity. Since \( v \) is constant for all three particles, their wavelengths are inversely proportional to their masses:

1. \( \lambda_e = \frac{h}{m_e v} \)

2. \( \lambda_p = \frac{h}{m_p v} \)

3. \( \lambda_d = \frac{h}{m_d v} \)

Here, \( m_e, m_p, \) and \( m_d \) denote the masses of the electron, proton, and deuteron, respectively. The following mass relationships are given:

- The electron's mass (\( m_e \)) is the smallest.

- The proton's mass (\( m_p \)) exceeds the electron's mass.

- The deuteron's mass (\( m_d \)), approximately double the proton's mass, is the largest.

Consequently, given the mass ordering \( m_e<m_p<m_d \), the wavelengths are ordered as \( \lambda_e>\lambda_p>\lambda_d \). Thus, the definitive relationship between the wavelengths is:

\( \lambda_e>\lambda_p>\lambda_d \)