# Calculating time taken for a set percentage of a reaction to occur based on half-life

The half-life of a first order homogenous gaseous reaction: $\ce{SO2Cl2 -> SO2 + Cl2}$ is 8.0 minutes. How long will it take for the concentration of $\ce{SO2Cl2}$ to be reduced to 1% of the initial value?

I solved it this way:

Let initial mass be 100, so that the final mass is 1. $$100 \cdot (\frac{1}{2})^n=1$$ I solve this for n, and multiply it by 8.0 to get the time taken.

Answer is spot on, and the method is identical to the one used in half life questions in Physics. Yet my Prof insists that I solve for the rate constant first, and then find half-life from that, and that my answer is wrong because n, the value for number of half-life periods, cannot be a decimal value.

The questions I found online all explicitly ask for the rate constant, which the question I got doesn't ask for.

Is my method right/wrong? Can you guys provide proof?

• See if you can derive your equation algebraically by working with the rate constant approach. – Chet Miller May 18 '16 at 0:29
• I know that approach, I'm concerned specifically with this method. I usually stick with methods I know are impossible for me to screw up, so I use this method. I just want to know what change is there to half life in chem that makes my method wrong. – Arv May 18 '16 at 1:18

Ryan showed that the rate constant is given by $k=\frac{ln(2)}{t_{1/2}}$. So the concentration at time t is given by $$C=C_0\exp(-kt)=C_0e^{-ln(2)\frac{t}{t_{1/2}}}$$But, $e^{-ln(2)}=\frac{1}{2}$. Therefore, $$C=C_0\left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}}$$Substituting $n=\frac{t}{t_{1/2}}$ gives $$C=C_0\left(\frac{1}{2}\right)^n$$
• @Beerhunter Who said it was an integer? It would be a non-integer. So what! Are you actually asking me how to determine the value of t in this specific problem when $C = 0.75C_0$ using this equation? – Chet Miller May 18 '16 at 22:53