If `r` is the interatomic distance, `a` and `b` are positive constants, `U` denotes potential energy which is a function dependent on `r` as follows :
`U=(a)/(r^(10) )-(b)/(r^(5))`.
The equilibrium distance between two atoms is

Options

A. `((b)/(2a))^(1//5)`
B. `((2a)/(b))^(1//5)`
C. `((b)/(2a))^(1/10)`
D. `((2a)/(b))^(1/10)`.

Explanation

To find the equilibrium distance between two atoms given the potential energy function \( U(r) = \frac{a}{r^{10}} - \frac{b}{r^{5}} \), we need to follow these steps:

### Step 1: Understand the condition for equilibrium
At equilibrium, the net force acting on the atoms is zero. The force can be derived from the potential energy function. The relationship is given by:
\[ F = -\frac{dU}{dr} \]
At equilibrium, we set this force to zero:
\[ -\frac{dU}{dr} = 0 \implies \frac{dU}{dr} = 0 \]

### Step 2: Differentiate the potential energy function
We need to differentiate \( U(r) \) with respect to \( r \):
\[ U(r) = \frac{a}{r^{10}} - \frac{b}{r^{5}} \]
Using the power rule for differentiation:
\[ \frac{dU}{dr} = -10 \cdot \frac{a}{r^{11}} + 5 \cdot \frac{b}{r^{6}} \]

### Step 3: Set the derivative equal to zero
Now, we set the derivative equal to zero to find the equilibrium condition:
\[ -10 \cdot \frac{a}{r^{11}} + 5 \cdot \frac{b}{r^{6}} = 0 \]

### Step 4: Rearranging the equation
Rearranging the equation gives:
\[ 5 \cdot \frac{b}{r^{6}} = 10 \cdot \frac{a}{r^{11}} \]

### Step 5: Cross-multiply to simplify
Cross-multiplying leads to:
\[ 5b \cdot r^{11} = 10a \cdot r^{6} \]

### Step 6: Simplify the equation
Dividing both sides by \( r^{6} \) (assuming \( r \neq 0 \)):
\[ 5b \cdot r^{5} = 10a \]
Now, divide both sides by 5:
\[ b \cdot r^{5} = 2a \]

### Step 7: Solve for \( r \)
Now, we can solve for \( r \):
\[ r^{5} = \frac{2a}{b} \]
Taking the fifth root of both sides gives:
\[ r = \left( \frac{2a}{b} \right)^{\frac{1}{5}} \]

### Final Answer
Thus, the equilibrium distance \( r \) between the two atoms is:
\[ r = \left( \frac{2a}{b} \right)^{\frac{1}{5}} \]

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