Question

One possible form for the potential energy (U) of a diatomic molecule (Fig. 40–8) is called the Morse Potential:

$U = U_{0} {[1 - e^{- a (r - r_{0})}]}^{2}$

(a) Show that r₀ represents the equilibrium distance and U₀ the dissociation energy.

Accepted Answer

Step 1: Start by analyzing the Morse potential equation: $$ U = U_0 [1 - e^{-a(r - r_0)}]^2 . $$The goal is to show that $$ r_0 $$ represents the equilibrium distance and $$ U_0 $$ represents the dissociation energy. Recall that the equilibrium distance corresponds to the point where the force acting on the molecule is zero, which is equivalent to the minimum of the potential energy function. Step 2: To find the equilibrium distance, take the derivative of $$ U $$ with respect to $$ r $$ and set it equal to zero: $$ \frac{dU}{dr} = 0 . $$Use the chain rule to differentiate $$ U $$ with respect to $$ r . $$The derivative of $$ e^{-a(r - r_0)} $$ with respect to $$ r  is$$ $$ -a e^{-a(r - r_0)} . $$Step 3: Solve $$ \frac{dU}{dr} = 0  to $$find the value of $$ r $$ that minimizes $$ U . $$Substituting $$ r = r_0 $$ into the equation, you will find that the exponential term $$ e^{-a(r - r_0)} $$ becomes $$ e^0 = 1 $$, which simplifies the expression. This confirms that $$ r_0  is $$the equilibrium distance. Step 4: To show that $$ U_0 $$ represents the dissociation energy, evaluate the potential energy $$ U  at$$ $$ r = r_0 . $$Substituting $$ r = r_0 $$ into the Morse potential equation, the term $$ [1 - e^{-a(r - r_0)}]^2 $$ becomes $$ [1 - 1]^2 = 0 $$, so $$ U = 0  at$$ $$ r = r_0 . $$Now, consider the limit as $$ r 	o \infty $$: the exponential term $$ e^{-a(r - r_0)} 	o 0 $$, and $$ U 	o U_0 . $$This shows that $$ U_0  is $$the energy required to dissociate the molecule. Step 5: Summarize the findings: $$ r_0 $$ represents the equilibrium distance because it minimizes the potential energy, and $$ U_0 $$ represents the dissociation energy because it is the energy required to separate the atoms completely ($$ r 	o \infty ).$$

One possible form for the potential energy (U) of a diatomic molecule (Fig. 40–8) is called the Morse Potential:

$U = U_{0} {[1 - e^{- a (r - r_{0})}]}^{2}$

(a) Show that r₀ represents the equilibrium distance and U₀ the dissociation energy.

Verified video answer for a similar problem:

Key Concepts

Morse Potential

Equilibrium Distance (r₀)

Dissociation Energy (U₀)

One possible form for the potential energy (U) of a diatomic molecule (Fig. 40–8) is called the Morse Potential:U=U0[1−e−a(r−r0)]2U=U_0\(\left\)[1-e^{-a(r-r_0)}\(\right\)]^2(a) Show that r0 represents the equilibrium distance and U0 the dissociation energy.

Verified video answer for a similar problem:

Key Concepts

Morse Potential

Equilibrium Distance (r₀)

Dissociation Energy (U₀)

One possible form for the potential energy (U) of a diatomic molecule (Fig. 40–8) is called the Morse Potential:

$U = U_{0} {[1 - e^{- a (r - r_{0})}]}^{2}$

(a) Show that r₀ represents the equilibrium distance and U₀ the dissociation energy.