The half-life of the first order reaction, A to products, is 53.2 s. What was the original concentration, (A)0 in moles per liter, if after 1.8 minutes, the concentration of A is 0.0783 moles per liter?

Not sure whether to use integrated rate law or half life law?

k = 0.693/t1/2

Substitute into the expression below for k.
ln(No/N) = kt.
No = unknown
N = 0.0783
t = time. You must change this t to seconds if you use the half life in seconds or you can change the half life in seconds to minutes in which case you make use t in the lower equation in minutes.

Thank you!

To determine whether to use the integrated rate law or the half-life law, we need to consider the information provided in the question.

Given that the half-life of the first-order reaction is 53.2 s, we can use this information to determine the original concentration, (A)0.

The half-life law for a first-order reaction states that the concentration of the reactant decreases by half after each half-life of the reaction.

Since 1.8 minutes = 1.8 * 60 = 108 seconds, we can calculate the number of half-lives that have elapsed by dividing the total time by the half-life:

Number of half-lives = Total time / Half-life = 108 s / 53.2 s = approximately 2.03

Now, we can use this information to determine the original concentration, (A)0, using the formula:

(A)t = (A)0 * (1/2)^n

where (A)t is the concentration of A after a certain time, (A)0 is the initial concentration, and n is the number of half-lives.

Substituting the given values, we have:

0.0783 M = (A)0 * (1/2)^2.03

To solve for (A)0, we need to isolate it on one side of the equation. Taking the natural logarithm (ln) of both sides can help with this:

ln(0.0783 M) = ln((A)0 * (1/2)^2.03)

Using the logarithmic property, ln(a * b) = ln(a) + ln(b), we can rewrite the equation as:

ln(0.0783 M) = ln((A)0) + ln((1/2)^2.03)

Now, we can rearrange the equation to solve for ln((A)0):

ln((A)0) = ln(0.0783 M) - ln((1/2)^2.03)

Finally, we can calculate (A)0 by taking the antilogarithm (exponentiation of e) of both sides:

(A)0 = e^(ln(0.0783 M) - ln((1/2)^2.03))

Using a calculator or a software program, evaluate this expression to find the original concentration, (A)0, in moles per liter.