The potential energy function for the force between two atoms in a diatomic molecule is approximately given by `U(x) =a/x^(12)-b/x^(6)` where a and b are constant and x is the distance between the atoms. Find the dissoociation energy of the molecule which is given as `D=[U(x- infty)-U_(at equilibrium)]`

Options

A. `(b^(2))/(2a)`
B. `(b^(2))/(12a)`
C. `(b^(2))/(4a)`
D. `(b^(2))/(6a)`

Answer

To find the dissociation energy \( D \) of the diatomic molecule given the potential energy function \( U(x) = \frac{a}{x^{12}} - \frac{b}{x^{6}} \), we will follow these steps:

### Step 1: Determine \( U(x \to \infty) \)

As \( x \) approaches infinity, the potential energy function \( U(x) \) can be evaluated:

\[ U(x \to \infty) = \frac{a}{\infty^{12}} - \frac{b}{\infty^{6}} = 0 - 0 = 0 \]

### Step 2: Find the equilibrium position

To find the equilibrium position, we need to set the force \( F \) to zero. The force is given by:

\[ F = -\frac{dU}{dx} \]

First, we differentiate \( U(x) \):

\[ U(x) = \frac{a}{x^{12}} - \frac{b}{x^{6}} \]

Calculating the derivative:

\[ \frac{dU}{dx} = -12 \frac{a}{x^{13}} + 6 \frac{b}{x^{7}} \]

Setting the force to zero:

\[ -12 \frac{a}{x^{13}} + 6 \frac{b}{x^{7}} = 0 \]

### Step 3: Solve for \( x \)

Rearranging the equation gives:

\[ 12 \frac{a}{x^{13}} = 6 \frac{b}{x^{7}} \]

Multiplying both sides by \( x^{13} \):

\[ 12a = 6b x^{6} \]

Dividing both sides by 6:

\[ 2a = b x^{6} \]

Thus, we can express \( x^{6} \):

\[ x^{6} = \frac{2a}{b} \]

### Step 4: Calculate \( U(x) \) at equilibrium

Now, we substitute \( x^{6} \) back into the potential energy function to find \( U \) at equilibrium:

\[ U(x) = \frac{a}{x^{12}} - \frac{b}{x^{6}} \]

Substituting \( x^{6} = \frac{2a}{b} \):

\[ x^{12} = \left(\frac{2a}{b}\right)^{2} = \frac{4a^{2}}{b^{2}} \]

Now substituting into \( U(x) \):

\[ U\left(\frac{2a}{b}\right) = \frac{a}{\frac{4a^{2}}{b^{2}}} - \frac{b}{\frac{2a}{b}} = \frac{b^{2}}{4a} - \frac{b^{2}}{2a} \]

Finding a common denominator:

\[ U\left(\frac{2a}{b}\right) = \frac{b^{2}}{4a} - \frac{2b^{2}}{4a} = -\frac{b^{2}}{4a} \]

### Step 5: Calculate the dissociation energy \( D \)

The dissociation energy \( D \) is given by:

\[ D = U(x \to \infty) - U(x_{\text{eq}}) \]

Substituting the values we found:

\[ D = 0 - \left(-\frac{b^{2}}{4a}\right) = \frac{b^{2}}{4a} \]

Thus, the dissociation energy of the molecule is:

\[ D = \frac{b^{2}}{4a} \]