There are four particles A, B, C and D with the same mass and equal kinetic energy K. If the kinetic energy of particle A becomes 4 K, B becomes 9 K, C becomes 64 K and D becomes 16 K, respectively, the fractional change in the de Broglie wavelength of the particles arranged in increasing order is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (B), (D), (C)
The de Broglie wavelength of a particle of mass \( m \) and kinetic energy \( K \) is
\( \displaystyle \lambda = \frac{h}{p} = \frac{h}{\sqrt{2mK}} \;\Rightarrow\; \lambda \propto \frac{1}{\sqrt{K}} \)
If the kinetic energy becomes \( nK \), the wavelength becomes \( \lambda/\sqrt{n} \). The wavelength decreases, and the size of the fractional change is
\( \displaystyle \frac{\lambda - \lambda/\sqrt{n}}{\lambda} = 1 - \frac{1}{\sqrt{n}} \)
\( A\ (n = 4):\ 1 - 1/2 = 0.50 \)
\( B\ (n = 9):\ 1 - 1/3 = 0.67 \)
\( C\ (n = 64):\ 1 - 1/8 = 0.875 \)
\( D\ (n = 16):\ 1 - 1/4 = 0.75 \)
The larger the increase in kinetic energy, the larger the fractional change, so the increasing order is A, B, D, C.