Potential energy is region is given by U= `alpha/r^(10)- beta/r^5` at equilibrium. Inter molecular distance between particle is given as r= `((2alpha)/beta)^(a/b)` . Then a will be:

Answer

To solve the problem, we need to find the value of \( a \) in the expression for the intermolecular distance \( r \) given by:

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

where \( U = \frac{\alpha}{r^{10}} - \frac{\beta}{r^5} \) is the potential energy.

### Step-by-Step Solution:

1. **Identify the Force from Potential Energy**:
The force \( F \) is related to the potential energy \( U \) by the equation:
\[ F = -\frac{dU}{dr} \]
We need to differentiate \( U \) with respect to \( r \).

2. **Differentiate \( U \)**:
Given:
\[ U = \frac{\alpha}{r^{10}} - \frac{\beta}{r^5} \]
We differentiate \( U \):
\[ \frac{dU}{dr} = -10\frac{\alpha}{r^{11}} + 5\frac{\beta}{r^6} \]
Thus, the force becomes:
\[ F = -\left(-10\frac{\alpha}{r^{11}} + 5\frac{\beta}{r^6}\right) = 10\frac{\alpha}{r^{11}} - 5\frac{\beta}{r^6} \]

3. **Set the Force to Zero for Equilibrium**:
At equilibrium, the force \( F \) is zero:
\[ 10\frac{\alpha}{r^{11}} - 5\frac{\beta}{r^6} = 0 \]
Rearranging gives:
\[ 10\frac{\alpha}{r^{11}} = 5\frac{\beta}{r^6} \]

4. **Simplify the Equation**:
Dividing both sides by 5:
\[ 2\frac{\alpha}{r^{11}} = \frac{\beta}{r^6} \]
Cross-multiplying gives:
\[ 2\alpha r^6 = \beta r^{11} \]

5. **Rearranging for \( r \)**:
Rearranging this equation:
\[ r^{11} = \frac{2\alpha}{\beta} r^6 \]
Dividing both sides by \( r^6 \) (assuming \( r \neq 0 \)):
\[ r^5 = \frac{2\alpha}{\beta} \]

6. **Finding \( r \)**:
Taking the fifth root:
\[ r = \left(\frac{2\alpha}{\beta}\right)^{\frac{1}{5}} \]

7. **Comparing with Given Expression**:
We are given:
\[ r = \left(\frac{2\alpha}{\beta}\right)^{\frac{a}{b}} \]
From the previous step, we have:
\[ r = \left(\frac{2\alpha}{\beta}\right)^{\frac{1}{5}} \]
Therefore, we can equate the exponents:
\[ \frac{a}{b} = \frac{1}{5} \]

8. **Finding \( a \)**:
Since \( b = 5 \), we can substitute:
\[ \frac{a}{5} = \frac{1}{5} \]
Thus, multiplying both sides by 5 gives:
\[ a = 1 \]

### Final Answer:
The value of \( a \) is \( 1 \).