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Unformatted text preview: he spring constant. Problem
46. In ionic solids such as NaCI (salt), the potential energy of a pair of ions takes the form U = br-n - ar-I, where r is the separation of the ions. For NaCI, a and b have the 81![values 4.04x 10-28 and 5.52x 10-98, respectively, and n = 8.22. Find the equilibrium sepa~ation between ions in NaC!' Solution
(a) The natural, or unstretched, length of the spring is the value of x for which the spring force is zero. Thus, Fx -dUfdx -2ax + b 0, when x bj2a = (3.12 N)j2(5.2 Njm) 30 em. (b) Hoo~'s Law defines the spring constant in terms of the stretch (in this case x - b/2a). Since Fi/; -2a(x - bj2a), the spring constant is k = 2a = 10.4 Njm. (Alternatively, k= -dFxjdx = d2Ujdx2.) = = Solution
At the equilibrium separation, the potential energy is . . 0 -(1.+1) + ar -2 a mimmum, or (dUjd r )eq -n breq' eq. Then req == (nbja)l/(n-l) = (8.22x5.52xlO-98+ 4.04xlO-28)1/7.22 (81 units) =2.82xlO-,10 m = 2.82 A. (The angstrom is a common non-81 unit of length, uSed in chemistry and atomic p...
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