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Chemistry 1Bkey - Chemistry 1b Solutions to the First...

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Chemistry 1b, Solutions to the First Midterm Examination 1. Atoms and ions can be excited to very high states called Rydberg states. Consider a He+ ion which has been excited to the n = 45 state. The nucleus of the one-electron ion is an alpha particle, e.g. 4 He. o Calculate the ionization energy of the ion. The helium cation is a one-electron species so the Bohr formula for the energy, E = - R(Z 2 /n 2 ), is exact. IE(eV) = -E = (13.6 eV)(2 2 /45 2 ) = 0.027 eV o Calculate the average size of the ion. r = (0.529 Å)(45 2 /2) = 540 Å o The result calculated above is an estimate of the uncertainty of the position of the electron in the ion. Use the result to estimate the uncertainty of the electron's momentum. Use the Heisenberg Uncertainty Principle. r p = (h/2 π ) p = h/2 π∆ r = (6.63 x 10 -34 J-s)/(2)(3.14)(5.4 x 10 -8 m) = 2.0 x 10 -27 Kg-m/s o Calculate the maximum possible value of the orbital angular momentum in units of h/ 2p . The quantum number l ranges from 0 to n-1 so the maximum value of l is 45-1 or 44. In units of h/2 π , L equals the square root of l(l+1), i.e. [44(45)] 0.5 o The ion is moving with a speed of 2.0 km/s. Calculate the de Broglie wavelength of the ion. λ = h/p = h/mv. We require the mass of the 4He ion in Kg. m = 0.0040 Kg/6.02 x 10 23 = 6.64 x 10 -27 Kg λ = (6.63 x 10 -27 J-s)/(6.64 x 10 -27 Kg)(2000 m/s) = 5.0 x 10 -11 m o Suppose that a sample of 1000 He + ions in the n = 45 state is prepared. When the excited ions decay to the ground state, will monochromatic or polychromatic radiation be emitted? Explain.
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