Math Snap

STEP 1

What is this asking?
How long does it take for half of the element's volume to disappear?
Watch out!
Don't mix up the remaining volume with the vaporized volume!

STEP 2

1. Set up the exponential decay formula
2. Plug in the given values
3. Solve for the half-life

STEP 3

We're dealing with exponential decay, so let's use the formula $V(t) = V_0 \cdot (1 - r)^t$, where $V(t)$ is the volume at time $t$, $V_0$ is the initial volume, $r$ is the rate of decay, and $t$ is the time.
Remember, this formula tells us how much volume is left, not how much has vaporized!

STEP 4

We're given the time in minutes, so we'll express the rate in "amount vaporized per minute."

STEP 5

We know the initial volume $V_0 = 50$ mL.

STEP 6

After 12 minutes, 12 mL vaporized, so the remaining volume is $50 - 12 = 38$ mL.
So, $V(12) = 38$.

STEP 7

The element vaporized 12 mL in 12 minutes, so the rate of decay per minute is $\frac{12}{50 \cdot 12} = \frac{1}{50}$.
Notice how we divide by the initial volume to get the fraction of the volume that vaporized per minute.

STEP 8

Now, plug everything into our formula: $38 = 50 \cdot (1 - \frac{1}{50})^{12}$.

STEP 9

We want to find the half-life, which is the time it takes for half the initial volume to vaporize.
So we set $V(t)$ to half of the initial volume: $\frac{50}{2} = 50 \cdot (1 - \frac{1}{50})^t$.

STEP 10

Divide both sides by 50: $\frac{1}{2} = (1 - \frac{1}{50})^t$ or $\frac{1}{2} = (\frac{49}{50})^t$.

STEP 11

Take the logarithm of both sides (you can use any base, but let's use base 10): $\log(\frac{1}{2}) = t \cdot \log(\frac{49}{50})$.

STEP 12

Now, solve for $t$: $t = \frac{\log(\frac{1}{2})}{\log(\frac{49}{50})}$.

STEP 13

Calculate the value of $t$: $t \approx \frac{-0.301}{-0.00877} \approx 34.32$.
So, the half-life is approximately 34.32 minutes.

The half-life of the element is approximately 34.32 minutes.