The potential energy (U) of diatomic molecule is a function dependent on r (interatomic distance )as `U=(alpha)/(r^(10))-(beta)/(r^(5))-3`
Where `alpha and beta` are positive constants. The equilrium distance between two atoms will be `((2alpha)/(beta))^((a)/(b))` where a=___

Answer

To solve the problem, we need to find the equilibrium distance \( r \) between two atoms based on the given potential energy function:

\[ U(r) = \frac{\alpha}{r^{10}} - \frac{\beta}{r^{5}} - 3 \]

### Step 1: Find the Force
The force \( F \) between the two atoms can be derived from the potential energy function using the relation:

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

### Step 2: Differentiate the Potential Energy
Now, we differentiate \( U(r) \) with respect to \( r \):

\[ \frac{dU}{dr} = -\frac{d}{dr}\left(\frac{\alpha}{r^{10}}\right) + \frac{d}{dr}\left(\frac{\beta}{r^{5}}\right) \]

Using the power rule for differentiation:

\[ \frac{d}{dr}\left(\frac{\alpha}{r^{10}}\right) = -10\frac{\alpha}{r^{11}}, \quad \frac{d}{dr}\left(\frac{\beta}{r^{5}}\right) = -5\frac{\beta}{r^{6}} \]

Thus,

\[ \frac{dU}{dr} = \frac{10\alpha}{r^{11}} - \frac{5\beta}{r^{6}} \]

### Step 3: Set the Force to Zero for Equilibrium
At equilibrium, the force \( F \) must be zero:

\[ -\frac{dU}{dr} = 0 \implies \frac{10\alpha}{r^{11}} - \frac{5\beta}{r^{6}} = 0 \]

### Step 4: Rearranging the Equation
Rearranging the equation gives:

\[ \frac{10\alpha}{r^{11}} = \frac{5\beta}{r^{6}} \]

Cross-multiplying yields:

\[ 10\alpha r^{6} = 5\beta r^{11} \]

### Step 5: Simplifying the Equation
We can simplify this equation:

\[ 2\alpha = \beta r^{5} \]

### Step 6: Solve for \( r \)
Now, we can solve for \( r \):

\[ r^{5} = \frac{2\alpha}{\beta} \]

Taking the fifth root of both sides:

\[ r = \left(\frac{2\alpha}{\beta}\right)^{\frac{1}{5}} \]

### Final Result
Thus, the equilibrium distance \( r \) can be expressed as:

\[ r = \left(\frac{2\alpha}{\beta}\right)^{\frac{1}{5}} \]

In the form given in the question, we have:

\[ r = \left(\frac{2\alpha}{\beta}\right)^{\frac{a}{b}} \]

Comparing this with our derived equation, we find \( a = 1 \) and \( b = 5 \).

### Answer
Thus, \( a = 1 \).

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