Question 96: Dual Nature of Matter and Radiation

Question

The de Broglie wavelengths of a proton and an $\alpha$ particle are $\lambda$ and $2 \lambda$ respectively. The ratio of the velocities of proton and $\alpha$ particle will be :

Accepted Answer

To find the ratio of velocities of two particles based on their de Broglie wavelengths, we can use the de Broglie wavelength formula, which relates the momentum of a particle to its wavelength. The de Broglie's wavelength formula is given by:
$$ \lambda = \frac{h}{p} $$
where:
$\lambda$ is the de Broglie wavelength,
$h$ is the Planck constant, and
$p$ is the momentum of the particle.
The momentum $p$ of a particle can also be expressed as the product of its mass $m$ and velocity $v$:
$$ p = mv $$
So, we can rewrite the de Broglie wavelength equation in terms of mass and velocity:
$$ \lambda = \frac{h}{mv} $$
For the proton (let's use subscript $p$ for proton), the wavelength is $\lambda$:
$$ \lambda_p = \frac{h}{m_p v_p}$$
For the $\alpha$ particle (let's use subscript $\alpha$ for alpha), the wavelength is $2\lambda$:
$$ 2\lambda = \frac{h}{m_\alpha v_\alpha}$$
We are interested in finding the ratio of the velocities $\frac{v_p}{v_\alpha}$. Using the given data about wavelengths:
$$ \lambda_p = \lambda $$
$$ \lambda_\alpha = 2\lambda $$
Using the de Broglie equation for both particles:
$$ \frac{h}{m_p v_p} = \lambda $$
$$ \frac{h}{m_\alpha v_\alpha} = 2\lambda $$
Dividing the second equation by the first equation gives us:
$$ \frac{\frac{h}{m_\alpha v_\alpha}}{\frac{h}{m_p v_p}} = \frac{2\lambda}{\lambda} $$
$$ \frac{m_p v_p}{m_\alpha v_\alpha} = \frac{2}{1} $$
$$ \frac{v_p}{v_\alpha} = \frac{2m_\alpha}{m_p} $$
Since we know an $\alpha$ particle consists of 2 protons and 2 neutrons (essentially four nucleons), the mass of an $\alpha$ particle is roughly four times the mass of a proton ($m_\alpha \approx 4m_p$).
Substituting $m_\alpha$ with $4m_p$ in the equation:
$$ \frac{v_p}{v_\alpha} = \frac{2(4m_p)}{m_p} $$
$$ \frac{v_p}{v_\alpha} = 2 \cdot 4 $$
$$ \frac{v_p}{v_\alpha} = 8 $$
Therefore, the ratio of the velocities of proton to $\alpha$ particle is $8:1$, which corresponds to Option A.

Answer

$8: 1$

Answer

$1: 2$

Answer

$1: 8$

Answer

$4: 1$

Dual Nature of Matter and Radiation

The de Broglie wavelengths of a proton and an \alpha particle are \lambda and 2 \lambda respectively. The ratio of the velocities of proton and \alpha particle will be :

Choose the correct answer: