Consider two separate ideal gases of electrons and protons having same number of particles. The temperature of both the gases are same. The ratio of the uncertainty in determining the position of an electron to that of a proton is proportional to :

Question

Choose correct option :
(A) Number of photons required for a light beam of 2000 pm wavelength and 1 Joule energy is \(1.01 \times 10^{16}\).
(B) Light with wavelength 300 nm has energy \(E_1\) and for wavelength 900 nm has energy \(E_2\), then \(E_1/E_2 = 1/3\).
(C) Frequency of light is \(4.5 \times 10^{16}\) Hz then its wavelength is \(6.7 \times 10^{-9}\) m.
(D) If electrons and protons are accelerated by same potential difference, then their de-Broglie wavelength are equal.

Answer 1

To solve this problem, we are going to examine the uncertainty in determining the position of two ideal gases made up of electrons and protons. Both gases have the same number of particles and are at the same temperature. According to Heisenberg's Uncertainty Principle, the product of the uncertainties in position and momentum for a particle is given by:

\(\Delta x \Delta p \geq \frac{\hbar}{2}\)

where \(\Delta x\) is the uncertainty in position, \(\Delta p\) is the uncertainty in momentum, and \(\hbar\) is the reduced Planck's constant.

Since both gases are at the same temperature, their average kinetic energies are the same, and thus:

\(\frac{1}{2} m_e v_e^2 = \frac{1}{2} m_p v_p^2 = \frac{3}{2} kT\)

where \(m_e\) and \(m_p\) are the masses of an electron and a proton respectively, and \(v_e\) and \(v_p\) are their velocities. This expression shows that:

\(\frac{v_e^2}{v_p^2} = \frac{m_p}{m_e}\)

From momentum \((p = mv)\), the uncertainty in momentum for electron and proton is given by:

\(\Delta p_e = m_e \Delta v_e\) and \(\Delta p_p = m_p \Delta v_p\)

The uncertainty principle can be rewritten in terms of velocities:

\(\Delta x \Delta v \geq \frac{\hbar}{2m}\)

We know that \(\Delta v_e \propto \sqrt{\frac{1}{m_e}}\) and \(\Delta v_p \propto \sqrt{\frac{1}{m_p}}\) because of similar kinetic energies (for both gases).

Using the uncertainty relation \(\Delta x \Delta v\) relation for both:

The ratio of the uncertainty in position of electron to that of proton:

\(\frac{\Delta x_e}{\Delta x_p} = \frac{\Delta v_p \cdot m_p}{\Delta v_e \cdot m_e}\)

Substitute the proportionality relations:

\(\frac{\Delta x_e}{\Delta x_p} = \sqrt{\frac{m_e}{m_p}} \cdot \frac{m_p}{m_e} = \sqrt{\frac{m_p}{m_e}}\)

So, the ratio of the uncertainty in determining the position of an electron to that of a proton is proportional to:

Answer: \(\sqrt{\frac{m_p}{m_e}}\)

Answer 2