The Reduced Mass

FIGURE 11.6 Two masses, mj and m2, are moving back and forth with respect to each other with the center of mass (CoM) unmoving. This circumstance is used to define the reduced mass /x. 

The negative sign indicates that the masses are moving in the opposite directions. By adding the mixed term ni2 dxi/dt) to both sides, we get 

It is very convenient in many cases to define relative coordinates instead of absolute coordinates. For example, specifying certain values of Cartesian coordinates is a way of using absolute coordinates. However, differences in Cartesian coordinates are relative, because the difference doesn t depend on the starting and ending values (for example, the difference between 5 and 10 is the same as the difference between 125 and 130). If we define the relative coordinate q as 

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What this means is that the kinetic energy of the oscillator can be represented by the kinetic energy of a single mass moving back and forth, if that single mass has the reduced mass of the two masses in the original system. This allows us to treat the fwo-particle harmonic oscillator as a one-particle harmonic oscillator and use the same equations and expressions that we derived for a simple harmonic oscillator. So all of the equations of the previous sections apply, assuming one uses the reduced mass of the system. For example, equation 11.3 becomes 

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To detemiine k E) from equation (A3.12.9) it is assumed that transition states with positivefomi products. Notmg that / f = p dqf/dt, where p is the reduced mass of the separating fragments, all transition states that lie within and + dq with positive will cross the transition state toward products in the time interval dt = pj dqf p. Inserting this expression into equation (A3.12.9), one finds that the reactant-to-product rate (i.e. flux) through the transition state for momenPim p is... 

Since the reduced mass depends only on the masses of the atoms (1/p = + Mm for a diatomic), which... 

If we think about two masses connected by a spring, each vibrating with respect to a stationary center of mass Xc of the system, we should expect the situation to be vei similar in form to one mass oscillating from a fixed point. Indeed it is, with only the substitution of the reduced mass p for the mass m... 

The hamionic oscillator of two masses is a model of a vibrating diatomic molecule. We ask the question, What would the vibrational frequency be for H2 if it were a hamionic oscillator The reduced mass of the hydrogen molecule is... 

The Hamiltonian in this problem contains only the kinetic energy of rotation no potential energy is present because the molecule is undergoing unhindered "free rotation". The angles 0 and (j) describe the orientation of the diatomic molecule s axis relative to a laboratory-fixed coordinate system, and p is the reduced mass of the diatomic molecule p=mim2/(mi+m2). 

The second term on the right-hand side is the coulombic potential energy for the attraction between charges —e and +e a distance r apart. The first term contains the reduced mass /r, equal to - - m ), for the system of an electron of mass and a proton of mass m. ... 

The dissociation energy is unaffected by isotopic substitution because the potential energy curve, and therefore the force constant, is not affected by the number of neutrons in the nucleus. However, the vibrational energy levels are changed by the mass dependence of 03 (proportional to where /r is the reduced mass) resulting in Dq being isotope-... 

D is the chemical energy of dissociation which cair be obtained from thermodynamic data, aird is the reduced mass of the diatomic molecule... 

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