JEE Advance - Chemistry (2004 - No. 8)
(a) The Schrodinger wave equation for hydrogen atom is
$$$\psi = {1 \over {4\sqrt {2\pi } }}{\left( {{1 \over {{a_0}}}} \right)^{3/2}}\left( {2 - {{{r_0}} \over {{a_0}}}} \right){e^{ - {r_0}/{a_0}}}$$$
Where a0 is Bohr's radius. If the radial node in 2s be at r0, then find r0 in terms of a0.
(b) A baseball having mass 100 g moves with velocity 100 m/s. Determine the value of wavelength of baseball.
$$$\psi = {1 \over {4\sqrt {2\pi } }}{\left( {{1 \over {{a_0}}}} \right)^{3/2}}\left( {2 - {{{r_0}} \over {{a_0}}}} \right){e^{ - {r_0}/{a_0}}}$$$
Where a0 is Bohr's radius. If the radial node in 2s be at r0, then find r0 in terms of a0.
(b) A baseball having mass 100 g moves with velocity 100 m/s. Determine the value of wavelength of baseball.
r0 = a0, λ = 6.626 × 10-34 m
r0 = 2a0, λ = 6.626 × 10-35 m
r0 = a0/2, λ = 6.626 × 10-36 m
r0 = 4a0, λ = 6.626 × 10-33 m
r0 = 2a0, λ = 6.626 × 10-34 m
Explanation
(a) Probability of finding electrons at any place = $$\psi _{2s}^2 = 0$$ at node
$$\therefore$$ $$\psi _{2s}^2 = 0 = {1 \over {4\sqrt {2\pi } }}{\left( {{1 \over {{a_0}}}} \right)^3}\left( {2 - {r \over {{a_0}}}} \right) \times {e^{ - r/{a_0}}}$$
or $$\left( {2 - {r \over {{a_0}}}} \right) = 0$$ or $$2 = {r \over {{a_0}}}$$ or $$r = 2{a_0}$$
(b) $$\lambda = {h \over {mv}}$$
$$h = 6.626 \times {10^{ - 34}}\,Js$$, m = 100 g = 0.1 kg v = 100 ms$$-$$1
$$\therefore$$ $$\lambda = {{6.626 \times {{10}^{ - 34}}\,kg\,{m^2}\,{s^{ - 1}}} \over {0.1\,kg \times 100\,m{s^{ - 1}}}}$$
$$ = 6.626 \times {10^{ - 35}}\,m$$
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