Speed of electron in its Ist Bohr's orbit is given by \(2.18 \times 10^6 \mathrm{~ms}^{-1}\). If the time…
Speed of electron in its Ist Bohr's orbit is given by \(2.18 \times 10^6 \mathrm{~ms}^{-1}\). If the time period of electron in \(n\)th orbit is measured as 4.10 femto second, the value of \(n\) is
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Solution
Given, speed of the electron in 1st Bohr's orbit,
\(v_1=2.18 \times 10^6 \mathrm{~m} / \mathrm{s}\)
Time period of \(n\)th orbit,
\(T_n=4.10 \mathrm{f}-\mathrm{s}=4.1 \times 10^{-15} \mathrm{~s}\)
Radius of Bohr's first orbit, \(r_1=0.53 \times 10^{-10} \mathrm{~m}\)
\(\therefore\) Orbital period of electron in Bohr's first orbit,
\(\begin{aligned}
& T_1=\frac{2 \pi r_1}{v_1}=\frac{2 \times 314 \times 0.53 \times 10^{-10}}{2.18 \times 10^6} \\
& T_1=1.52 \times 10^{-16} \mathrm{~s}
\end{aligned}\)
Again, Time period of \(n\)th orbit is given by
\(\Rightarrow \quad \begin{aligned}
T_n & =n^3 T_1 \\
\Rightarrow \quad n^3 & =\frac{T_n}{T_1}=\frac{4.1 \times 10^{-15}}{1.52 \times 10^{-16}}=26.98 \\
n^3 & =27 \Rightarrow n=(27)^{1 / 3} \Rightarrow n=3
\end{aligned}\)