Problem Set 1 Solutions

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1 Chemistry 36 Dr. Jean M. Standard Problem Set Solutions. The first 4 lines in the visible region of atomic line spectrum of hydrogen atom occur at wavelengths of 656., 486., 434.0, and 40. nm (this is known as the Balmer series. Using Bohr s theory, show that these correspond to transitions in which the smaller integer in the equation is always n. Bohr s theory gives the following formula for the wavelengths in the atomic line spectrum of hydrogen, or λ $ me 4 ' $ % 8ε o h 3 c ( n % n λ R $ n ', % n ( ' ( where R is the Rydberg constant, cm. Note that in this equation n is always smaller than n (so that the wavelength is positive. Therefore, we would set n and try various integers greater than for the value of n. Substituting n and n 3, for example, leads to the result λ R $ ' % 3 ( ( cm $ 4 ' % 9 ( λ 54.3cm. Converting to wavelength by taking the inverse of both sides and converting to nm, we get λ 54.3cm % cm ' cm ( 07 nm λ 656.nm. Note that this value closely matches the first wavelength listed in the problem. ( *

2 . Continued We can now carry out similar calculations for n and n 4, 5, and 6 to see if these transitions match the other wavelengths listed in the problem. The results are summarized in the table below. n n Calculated wavelength (nm Experimental wavelength (nm As can be seen from the table, the calculated wavelengths are in good agreement with the experimental wavelengths for n and n 3, 4, 5, and 6.. The ionization potential of an atom is defined as the energy required to completely remove an electron. Using Bohr's theory, calculate the ionization potential of the hydrogen atom. The ionization potential (IP is the energy required to remove an electron from the first orbit (n to infinite separation (n, IP E n E n. From the Bohr theory, the energy levels of the hydrogen atom are E n m e 4 8ε o n h. Substituting the expression for the Bohr energies into the equation for the ionization potential yields IP IP m e4 8ε o h ' m e 4 8ε o h. % ( * Substituting for the constants and converting from Joules to Electron Volts gives IP ( kg ( C 4 ( ( J s C J - m J IP 3.6 ev. $ ev ' % J (

3 3. Determine the absorption frequency in wavenumbers (cm for an electron in the hydrogen atom undergoing a transition from the n level to the n5 level. From the Bohr theory, the energy levels of the hydrogen atom are E n The energy difference between the n and n5 levels is m e 4 8ε o n h. ΔE E 5 E m e 4 8ε o 5 h % m e 4 ( ' 8ε o h * m e 4 % 8ε o h ( ' 5 * ( kg C ( 4 ( ( J s C J - m - % 4 ( ' * 5 ΔE J. To determine the absorption frequency, the photon energy, E photon hν, must be set equal to the energy spacing between the levels, E photon hν ΔE. Solving for the frequency and substituting yields, ν ΔE h J J s ν s -. This result gives the frequency in s. To get the "frequency" in units of cm, the conversion is the speed of light, ν ( cm ( ν s - ( c cm/s s cm/s ν 3045 cm -. 3

4 4. Show that Bohr's condition for the quantization of angular momentum in the hydrogen atom, n! mvr, can be obtained by requiring an integer number of standing waves around an electron orbit of radius r. (Hint: you also need to use the expression for the debroglie wavelength. To start, note that the circumference of an orbit of radius r is π r. If we require an integer number of waves with wavelength λ to fit around the circumference of the orbit, this leads to the condition: π r nλ. For matter, the debroglie wavelength is λ h p h. If we substitute this expression into the equation above, mv we get This equation can be rearranged to yield π r nh mv. m v r nh π, which is the condition for quantization of angular momentum. 4

5 5. Data for the photoelectric effect experiment on sodium metal is given below. Use this data to determine a value for Planck's constant, h, and the work function φ for sodium metal. Report your result for h in units of Joule-seconds, and your value of φ in electron volts (ev. Photon Frequency Electron Kinetic Energy (s (ev From the equation KE hν φ, it is easy to see that a plot of KE on the y-axis vs. frequency on the x-axis should yield a straight line with slope h and intercept φ. The graph is shown in the figure below..5 KE (ev frequency (sec- 5

6 5. Continued From the fit of a linear trendline, the graph has a slope of ev s and an intercept of.63 ev. Therefore, using the conversion factor ev J, we get Planck's constant h from the slope: h J s (close to the literature value of J s. From the intercept (which is φ, we get the work function: φ.63 ev. Note that, if you did not plot the points, you could use any two of the points and solve two equations for two unknowns to get h and φ. Your numerical values will be slightly different in that case. 6. The threshold wavelength for the ejection of photoelectrons from sodium metal is 540 Å. Calculate the velocity of photoelectrons ejected by light of wavelength 4000 Å. The threshold frequency ν 0 is the frequency at which the ejected electrons have zero kinetic energy: KE 0 hν 0 φ, or ν 0 φ h. We can use this equation to determine the work function of sodium metal. However, we are given the threshold wavelength and the equation above contains the threshold frequency. Since λν c, we can solve to get the threshold wavelength as λ 0 c ν 0. Substituting the expression for the threshold frequency, the equation becomes λ 0 hc φ. Solving for the work function φ gives φ hc λ 0 ( J s ms m φ J. ( ( 6

7 6. Continued Now, to get the velocity of ejected photoelectrons at frequencies other than the threshold, we use the equation KE hν φ. Substituting ν c /λ for the frequency and KE mv for the kinetic energy, we get This equation can be solved for the velocity to yield KE hν φ mv hc λ φ. v + -, m % ' hc λ φ (. * 0 / /. Substituting λ 4000 Å ( or m and φ from above, and using the electron mass of m kg, the photoelectron velocity is v ( ( m/s ( * $, Js, kg % m + ' J / / (. / v m/s. 7

8 7. Using Bragg's Law, nλ d sinθ, determine the distance in Å between crystal planes in an atomic solid if electromagnetic radiation of frequency s incident at a 40º angle creates constructive interference fringes (assume n. Setting n and solving for the distance d yields d λ sinθ. Converting wavelength into frequency of light using the relation λ c /ν leads to the expression d c ν sinθ. Substituting, d c ν sinθ ( ms s ( sin( 40 o d m or 7.0 Å. 8

9 8. Calculate the debroglie wavelength of an electron in the first Bohr orbit in the hydrogen atom. Does quantum or classical mechanics apply to this electron? The debroglie wavelength is defined as λ h p h mv. The Bohr quantization condition for angular momentum is Solving the second equation for m v gives m v r n h π. m v nh π r. Substituting this into the debroglie wavelength expression leads to the relation λ πhr nh πr n. To get a numerical value for the debroglie wavelength, we use the expression from class for the quantized orbit, r ε on h π m e. Substituting this into the debroglie expression and using n for the lowest Bohr orbit leads to λ π n ε on h π m e ε o n h m e ( ( ( J s ( kg ( C C J - m - λ m or 3.3 Å. Since the debroglie wavelength is the same order of magnitude as the path that the electron travels (in this case, it equals the circumference of the orbit, π r, quantum mechanics should be applied to the electron. 9

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