How do I determine the value of the first-order rate constant using time?

You know the equation:

R = k[X]

For a reaction of first order.

k is given by:

k =\frac{R}{[X]}

Unless you have the value of the rate and the concentration, you cannot deduce the value of k. Time and concentration alone are not enough.

That is, if I understood your question properly as you didn't give much details.