The potential energy function for a diatomic molecule is `U(x) = (a)/(x^(12))-b/(x^6)` entered In stable equilibrium, the distance between the particles is

Options

A. `((2a)/b)^(1//6)`
B. `((a)/b)^(1//6)`
C. `((b)/(2a))^(1//6)`
D. `((b)/(a))^(1//6)`

Explanation

To find the distance between the particles in stable equilibrium for the given potential energy function \( U(x) = \frac{a}{x^{12}} - \frac{b}{x^6} \), we will follow these steps:

### Step 1: Understand the condition for stable equilibrium
In stable equilibrium, the net force acting on the particles must be zero. The force can be derived from the potential energy function as:
\[ F(x) = -\frac{dU}{dx} \]
Thus, for stable equilibrium, we need to set \( F(x) = 0 \).

### Step 2: Differentiate the potential energy function
We differentiate the potential energy function \( U(x) \) with respect to \( x \):
\[ U(x) = \frac{a}{x^{12}} - \frac{b}{x^6} \]
Differentiating \( U(x) \):
\[ \frac{dU}{dx} = -12 \frac{a}{x^{13}} + 6 \frac{b}{x^7} \]

### Step 3: Set the derivative equal to zero
To find the equilibrium position, we set the derivative equal to zero:
\[ -12 \frac{a}{x^{13}} + 6 \frac{b}{x^7} = 0 \]

### Step 4: Rearrange the equation
Rearranging the equation gives:
\[ 12 \frac{a}{x^{13}} = 6 \frac{b}{x^7} \]
This simplifies to:
\[ \frac{12a}{x^{13}} = \frac{6b}{x^7} \]

### Step 5: Cross-multiply and simplify
Cross-multiplying gives:
\[ 12a \cdot x^7 = 6b \cdot x^{13} \]
Dividing both sides by \( 6 \) results in:
\[ 2a \cdot x^7 = b \cdot x^{13} \]

### Step 6: Solve for \( x \)
Now, we can solve for \( x \):
\[ 2a = b \cdot x^{6} \]
Thus,
\[ x^6 = \frac{2a}{b} \]
Taking the sixth root gives:
\[ x = \left(\frac{2a}{b}\right)^{\frac{1}{6}} \]

### Final Answer
The distance between the particles in stable equilibrium is:
\[ x = \left(\frac{2a}{b}\right)^{\frac{1}{6}} \]

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