Homework Problem Set 5 Solutions ( )2

More documents

Recommendations

Info

3. Consider the potential energy function studied in problem 2 with the same parameters. Since the potential energy function described in problem 2 is harmonic, the bond undergoes simple harmonic motion as a function of time. The exact analytical solutions of Newton's equations for a harmonic oscillator with an initial position of zero and an initial velocity v(0) are 6 € x(t) = v(0) ω sin( ω t) v(t) = v(0) cos( ω t) . The parameter ω is the angular velocity and is defined by the relation ω = where m is the mass. The angular velocity ω is related to the harmonic frequency of oscillation ν o , € k m , € ν o = ω 2π . a.) Using the same parameters as € given in problem 2, determine the harmonic frequency of vibration for this system. For simple harmonic motion, the harmonic frequency of vibration ν o is ν o = = ω 2π = 1 2π 1 2π ν o = 2.52 ×10 13 s −1 . k m € ⎛⎛ 720 kcalmol −1 Å −2 ⎞⎞ ⎛⎛ 4.184 kJ ⎜⎜ ⎝⎝ 0.012 kg mol −1 ⎟⎟ ⎜⎜ ⎠⎠ ⎝⎝ 1kcal ⎞⎞ ⎛⎛ ⎟⎟ 1000 J ⎞⎞ ⎛⎛ 1Å ⎞⎞ ⎜⎜ ⎟⎟ ⎜⎜ ⎠⎠ ⎝⎝ 1kJ ⎠⎠ ⎝⎝ 10 −10 ⎟⎟ m⎠⎠ 2 €
3. continued 7 b.) Using the analytic solution to Newton's equations for the harmonic oscillator, plot the bond displacement coordinate x as a function of time. For simple harmonic motion and the provided initial conditions, the analytic expression for the position is x(t) = v(0) ω sin( ω t) . Using the harmonic vibrational frequency ν o calculated in part (a), the angular frequency ω can be determined, € € ω = 2π ν o ( ) = 2π 2.52 ×10 13 s −1 ω = 1.58 ×10 14 s −1 . The initial velocity is € v( 0) = 8400 m/s . Thus, the position as a function of time is given by the equation € ( ) = v(0) x t sin( ω t) ω ⎛⎛ 8400 m/s ⎞⎞ = ⎜⎜ ⎝⎝ 1.58 ×10 14 s −1 ⎟⎟ sin 1.58 ×10 14 s −1 t ⎠⎠ = 5.31×10 −11 m ( ) ( ) sin( 1.58 ×10 14 s −1 t) ( ) sin( 1.58 ×10 14 s −1 t) . x( t) = 0.531Å A plot of this function versus time is shown in the figure below. 0.6 € 0.4 0.2 x(t) (Å) 0.0 0.0 0.1 0.2 0.3 0.4 0.5 -0.2 -0.4 -0.6 time (ps)
Page 1 and 2: Chemistry 380.37 Dr. Jean M. Standa
Page 3 and 4: 1 d.) continued 3 To complete the d
Page 5: 2. continued 5 b.) What is the kine
Page 9 and 10: 4. Again consider the simple one-di
Page 11 and 12: 4 a.) continued 11 Averaging the se
Page 13 and 14: 5a, b.) continued 13 The velocity d
Page 15 and 16: 15 6. The equipartition theorem sta

Homework Problem Set 5 Solutions ( )2

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?