Question

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 uncertainity in determining the position of an electron to that of a proton is proportional to

• A. $\frac{m_p}{m_e}$
• B. $\sqrt{\frac{m_p}{m_e}}$
• C. $\sqrt{\frac{m_e}{m_p}}$
• D. ${\left[\frac{m_e}{m_p}\right]}^{\frac{3}{2}}$

Answer 1

The correct option is B $\sqrt{\frac{m_p}{m_e}}$

From uncertainity principle
$\Delta x\Delta p\ge \frac{h}{4\pi }$

The ratio of the uncertainity in determining the position of an electron to that of a proton is $\frac{\Delta x_e}{\Delta x_p}=\frac{\Delta p_p}{\Delta p_e}=\frac{m_p\Delta v_p}{m_e\Delta v_e}$

RMS speed of molecules of an ideal gas is given by $v=\sqrt{\frac{3RT}{m}}$

On substitution, we get
$\frac{\Delta x_e}{\Delta x_p}=\frac{m_p\sqrt{\frac{3RT}{m_p}}}{m_e\sqrt{\frac{3RT}{m_e}}}=\sqrt{\frac{m_p}{m_e}}$