Respuesta :
The expression for the bonding energy E₀ in terms of the equilibrium interionic separation r₀ and the given constants is;
E₀ = rD[(exp(-r₀/ρ) + exp(-r/ρ)] - EN/r₀
Bonding Energy and Net Potential Energy
We are given the expression;
EN = -C/r + Dexp(-r/ρ)
where;
EN is net potential energy
r is the interionic separation
C, D, and rho(ρ) are constants whose values depend on the specific material.
The formula for the bonding energy is usually;
E₀ = -C/r₀ + D exp(-r₀/ρ)
Step 1; We are to differentiate EN with respect to r. Thus, we have;
dEN/dr = C/r² - D exp(-r/p)
Step 2; We are to solve for C in terms of D, rho(ρ) and r₀. Thus;
E₀ + (C/r₀) = -D*exp(-r₀/ρ)
⇒ C/r₀ = -Dexp(-r₀/ρ) - E₀
Thus, multiplying both sides by r₀ gives;
C = -r₀(Dexp(-r₀/ρ) + E₀)
Step 3; We are to determine the expression for E₀ by substitution for C in the equation given. This gives us;
EN = -r₀(Dexp(-r₀/ρ) + E₀)/r + Dexp(-r/ρ)
EN = r₀*D*exp(-r₀/ρ) - (r₀E₀/r) + D*exp(-r/ρ)
EN + (r₀E₀/r) = r₀*D*exp(-r₀/ρ) + D*exp(-r/ρ)
r₀E₀/r = D[(exp(-r₀/ρ) + exp(-r/ρ)] - EN
Thus; E₀ = rD[(exp(-r₀/ρ) + exp(-r/ρ)] - EN/r₀
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