91Ó°ÊÓ

According to Bohr's theory, the energy of an electron in a hydrogen atom is given by: \(E = -\frac{13.6 eV}{n^2}\) Where n is the principal quantum number (n = 1, 2, 3, ...)

The linear momentum magnitude (P) can be calculated using the de Broglie wavelength: \(P = \frac{h}{\lambda}\) The circumference of the orbit is an integral multiple of the de Broglie wavelength: \(2 \pi r = n\lambda \) Combining the two equations, we get: \(P = \frac{h}{\lambda} = \frac{hn}{2 \pi r}\)

The expression for the orbital radius in terms of the principal quantum number (n) is given by: \(r = \frac{n^2h^2}{4\pi^2me^2}\) Where \(h\) is Planck's constant, \(m\) is the mass of the electron, and \(e\) is the elementary charge.

(A) Is \(EPr\) proportional to \(\frac{1}{n}\)? Using the derived expressions for \(E\), \(P\), and \(r\): \(EPr = \left(-\frac{13.6}{n^2}\right) \left(\frac{hn}{2\pi r}\right) \left(\frac{n^2h^2}{4\pi^2me^2}\right)\) Simplifying, all the terms containing \(n\) cancel out. Thus, \(EPr\) is not proportional to \(\frac{1}{n}\), and Statement A is False. (B) Is \(\frac{P}{E}\) proportional to \(n\)? Using the expressions for \(E\) and \(P\): \(\frac{P}{E} = \frac{\frac{hn}{2\pi r}}{-\frac{13.6}{n^2}}\) After plugging in the expression for \(r\), we find that all the terms containing \(n\) cancel out. Thus, \(\frac{P}{E}\) is not proportional to \(n\), and Statement B is False. (C) Is \(Er\) constant for all orbits? Using the derived expressions for \(E\) and \(r\): \(Er = \left(-\frac{13.6}{n^2}\right)\left(\frac{n^2h^2}{4\pi^2me^2}\right)\) Simplifying, all the terms containing \(n\) cancel out, resulting in a constant value. Thus, statement C is true. (D) Is \(Pr\) proportional to \(n\)? Using the derived expressions for \(P\) and \(r\): \(Pr = \left(\frac{hn}{2\pi r}\right)\left(\frac{n^2h^2}{4\pi^2me^2}\right)\) Simplifying, we find that \(Pr\) is proportional to \(n\). Thus, statement D is true.

According to Bohr's theory for hydrogen atom: (A) False: \(EPr\) is not proportional to \(\frac{1}{n}\) (B) False: \(\frac{P}{E}\) is not proportional to \(n\) (C) True: \(Er\) is constant for all orbits (D) True: \(Pr\) is proportional to \(n\)