# What is the half-life when the initial concentration is 0.51 M?

##### 1 Answer

The identity of the substance is not known (so you can't look up the half-life itself), nor is the total amount of time passed known, so this question can only be answered in general, with hypothetical numbers.

The **half-life equation** is:

#\mathbf([A] = 1/(2^(t"/"t_"1/2")) [A]_0)#

Even though a *first-order* half-life (general-chemistry half-life decay) technically doesn't depend on the *initial concentration* of the substance, we would still have to know either:

- The final concentration
**OR**the fraction of the substance leftover, **AND**the total time passed,

since the half-life equation contains the

initialandfinalconcentrations, as well as thetotaltime passed and thehalf-lifeitself. The rate itself doesn't change due to concentration, though.

From here, you wouldn't need to know the identity of the substance, but we'd need to make up some numbers.

If we wanted, one option is to know that the final concentration is, say,

Then we know what *fraction* of the substance was left after time

Again, we have no idea how much total time passed, but let's pretend

In this case, we have:

#1/([A]) = 2^(t"/"t_"1/2")/([A]_0)#

#([A]_0)/([A]) = 2^(t"/"t_"1/2")#

Now to take the base 2 logarithm and get the exponent out.

#log_2(([A]_0)/([A])) = t/(t_"1/2")#

#1/(log_2(([A]_0)/([A]))) = t_"1/2"/t#

#color(blue)(t_"1/2" = t/(log_2(([A]_0)/([A]))#

Now simply plug in your numbers:

#t = "1 hour"# #[A]_0 = "0.51 M"# #[A] = "0.1275 M"#

Note that you could rewrite