The van't Hoff factor for a solute that associates in solution is

Options

A. zero
B. 1
C. less than 1
D. more than one

Explanation

To determine the van't Hoff factor (i) for a solute that associates in solution, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Association**:
When a solute associates in solution, it means that multiple solute particles combine to form fewer particles. For example, if we have a solute AB that dissociates into n particles, then we can represent this as:
\[ AB \rightarrow n \text{ particles} \]

2. **Initial Moles of Solute**:
Let's assume we start with \( n \) moles of the solute AB.

3. **Degree of Association**:
Let \( \alpha \) be the degree of association, which represents the fraction of the solute that has associated.

4. **Calculating Remaining Moles**:
If \( \alpha \) moles of AB associate, then the remaining moles of AB will be:
\[ n(1 - \alpha) \]
This is because \( \alpha \) moles have reacted to form products.

5. **Total Moles After Association**:
The total number of moles after association will be the moles that remain plus the moles that have formed due to association. If \( \alpha \) moles of AB associate to form \( n \) moles of products, the total moles will be:
\[ (n(1 - \alpha)) + \left(\frac{\alpha}{n}\right) \]

6. **Van't Hoff Factor (i)**:
The van't Hoff factor \( i \) is defined as the ratio of the total number of particles in solution after dissociation to the number of formula units initially dissolved. Therefore, we can express \( i \) as:
\[ i = (1 - \alpha) + \frac{\alpha}{n} \]

7. **Simplifying the Expression**:
Rearranging the expression for \( i \):
\[ i = 1 - \alpha + \frac{\alpha}{n} \]
This can be further simplified to:
\[ i = 1 - \left(1 - \frac{1}{n}\right) \alpha \]

8. **Analyzing the Result**:
Since \( \alpha \) is a fraction (0 ≤ α ≤ 1), and \( n \) is greater than 1 for association, the term \( \left(1 - \frac{1}{n}\right) \) will be positive. Thus, \( i \) will be less than 1, indicating that the van't Hoff factor for a solute that associates in solution is less than 1.

### Conclusion:
The van't Hoff factor \( i \) for a solute that associates in solution is less than 1.

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