For the reaction `A to ` Product
`k=(1)/(t)[(1)/(a-x)-(1)/(a)]`
Where k is rate constant (in `mol^(-1)s^(-1)`) and `(a-x)` are the concentration of A at the start and after time interval t sec. A graph between `(1)/(a-x)` and time t is of the following

Options

A.
B.
C.
D.

Explanation

To solve the problem, we start with the given equation for the reaction:

\[ k = \frac{1}{t} \left( \frac{1}{a-x} - \frac{1}{a} \right) \]

Where:
- \( k \) is the rate constant (in \( \text{mol}^{-1}\text{s}^{-1} \)),
- \( a \) is the initial concentration of reactant A,
- \( x \) is the amount of A that has reacted after time \( t \).

### Step 1: Rearranging the Equation

We can rearrange the equation to express \( \frac{1}{a-x} \) in terms of \( t \):

\[ kt = \frac{1}{a-x} - \frac{1}{a} \]

Adding \( \frac{1}{a} \) to both sides gives:

\[ \frac{1}{a-x} = kt + \frac{1}{a} \]

### Step 2: Identifying the Linear Form

This equation can be compared to the linear equation \( y = mx + c \), where:
- \( y = \frac{1}{a-x} \),
- \( m = k \) (the slope),
- \( x = t \) (the independent variable),
- \( c = \frac{1}{a} \) (the y-intercept).

### Step 3: Analyzing the Graph

From the rearranged equation, we can conclude:
- The graph of \( \frac{1}{a-x} \) versus \( t \) is a straight line.
- The slope of this line is positive since \( k \) is a positive rate constant.
- The y-intercept is \( \frac{1}{a} \).

### Step 4: Evaluating the Options

Now, we need to evaluate the given options for the graph:
1. A graph with a negative slope is incorrect because \( k \) is positive.
2. A graph with a zero slope is incorrect because \( k \) is not zero.
3. A graph with a positive slope and a y-intercept of \( \frac{1}{a} \) is correct.
4. A graph with a zero slope is again incorrect.

Thus, the correct option is the one that shows a straight line with a positive slope and a y-intercept at \( \frac{1}{a} \).

### Conclusion

The graph of \( \frac{1}{a-x} \) versus time \( t \) will be a straight line with a positive slope \( k \) and a y-intercept of \( \frac{1}{a} \).