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Let \({\lambda }_{e},{\lambda }_{p}\) and \({\lambda }_{d}\) be the wavelengths associated with an electron, a proton an…

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Let \({\lambda }_{e},{\lambda }_{p}\) and \({\lambda }_{d}\) 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

\({\lambda }_{d}>{\lambda }_{p}>{\lambda }_{e}\)

b

\({\lambda }_{e}>{\lambda }_{p}>{\lambda }_{d}\)

c

\({\lambda }_{\mathrm{p}}>{\lambda }_{\mathrm{e}}>{\lambda }_{\mathrm{d}}\)

d

\({\lambda }_{e}={\lambda }_{\mathrm{p}}={\lambda }_{\mathrm{d}}\)

✓ Correct answer: b)

\({\lambda }_{e}>{\lambda }_{p}>{\lambda }_{d}\)

Explanation

Wavelengths associated with an electron ( \(\lambda_e\) ), proton ( \(\lambda_p\) ), and deuteron ( \(\lambda_d\) ), all moving with the same speed.

Using de Broglie's equation: \(\boldsymbol{\lambda}=\frac{\boldsymbol{h}}{\boldsymbol{m} v}\)
- Since all particles have the same speed (v), their wavelengths are inversely proportional to their masses.
- Electron has the smallest mass, so it has the longest wavelength.
- Deuteron has the highest mass, so it has the shortest wavelength.

Mass relation: \(m_e<m_p<m_d\)
thus,
wavelength relation: \(\lambda_e>\lambda_p>\lambda_d\)

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