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The two particles A and B have de Broglie wavelengths 1 nm and 5 nm respectively. If mass of A is four times the mass of B, the ratio of kinetic energies would be
- a)5 : 1
- b)25 : 4
- c)20 : 1
- d)5 : 4
Correct answer is option 'B'. Can you explain this answer?
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The two particles A and B have de Broglie wavelengths 1 nm and 5 nm re...
Solution:
Given, $\lambda_A = 1 nm$, $\lambda_B = 5 nm$ and $m_A = 4m_B$
We know that the de Broglie wavelength of a particle is given by $\lambda = \dfrac{h}{mv}$, where $h$ is the Planck's constant, $m$ is the mass of the particle and $v$ is the velocity of the particle.
Thus, we can write the ratio of velocities of particles A and B as:
$\dfrac{v_A}{v_B} = \dfrac{\lambda_B}{\lambda_A} = \dfrac{5 nm}{1 nm} = 5$
Now, we can write the ratio of kinetic energies as:
$\dfrac{K_A}{K_B} = \dfrac{1}{2}m_Av_A^2 \div \dfrac{1}{2}m_Bv_B^2$
$\dfrac{K_A}{K_B} = \dfrac{m_A}{m_B} \times \left(\dfrac{v_A}{v_B}\right)^2$
Substituting the values, we get:
$\dfrac{K_A}{K_B} = \dfrac{4}{1} \times 5^2 = 100$
Therefore, the ratio of kinetic energies of particles A and B is 25 : 4. Hence, option (b) is the correct answer.
Given, $\lambda_A = 1 nm$, $\lambda_B = 5 nm$ and $m_A = 4m_B$
We know that the de Broglie wavelength of a particle is given by $\lambda = \dfrac{h}{mv}$, where $h$ is the Planck's constant, $m$ is the mass of the particle and $v$ is the velocity of the particle.
Thus, we can write the ratio of velocities of particles A and B as:
$\dfrac{v_A}{v_B} = \dfrac{\lambda_B}{\lambda_A} = \dfrac{5 nm}{1 nm} = 5$
Now, we can write the ratio of kinetic energies as:
$\dfrac{K_A}{K_B} = \dfrac{1}{2}m_Av_A^2 \div \dfrac{1}{2}m_Bv_B^2$
$\dfrac{K_A}{K_B} = \dfrac{m_A}{m_B} \times \left(\dfrac{v_A}{v_B}\right)^2$
Substituting the values, we get:
$\dfrac{K_A}{K_B} = \dfrac{4}{1} \times 5^2 = 100$
Therefore, the ratio of kinetic energies of particles A and B is 25 : 4. Hence, option (b) is the correct answer.
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The two particles A and B have de Broglie wavelengths 1 nm and 5 nm respectively. If mass of A is four times the mass of B, the ratio of kinetic energies would bea)5 : 1b)25 : 4c)20 : 1d)5 : 4Correct answer is option 'B'. Can you explain this answer?
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