Calculate the molar absorptivity of KMnO4 given its concentration in ppm and absorbance.

The molar mass of potassium (K) is $39.098 \text{ g/mol}$.
The molar mass of manganese (Mn) is $54.938 \text{ g/mol}$.
The molar mass of oxygen (O) is $15.999 \text{ g/mol}$.

$$\text{Molar mass of KMnO}_4 = 39.098 + 54.938 + (4 \times 15.999) = 158.032 \text{ g/mol}$$

A concentration of $25 \text{ ppm}$ means $25 \text{ mg}$ of KMnO$_4$ per liter of solution.
First, convert milligrams to grams:

$$25 \text{ mg} = 0.025 \text{ g}$$

Now, calculate the moles of KMnO$_4$ in $1 \text{ L}$ of solution:

$$\text{Moles of KMnO}_4 = \frac{\text{Mass}}{\text{Molar mass}} = \frac{0.025 \text{ g}}{158.032 \text{ g/mol}} \approx 1.58195 \times 10^{-4} \text{ mol}$$

The molar concentration ($C$) is:

$$C = \frac{1.58195 \times 10^{-4} \text{ mol}}{1 \text{ L}} = 1.58195 \times 10^{-4} \text{ M}$$

The Beer-Lambert Law is given by $A = \epsilon C L$, where $A$ is absorbance, $\epsilon$ is molar absorptivity, $C$ is molar concentration, and $L$ is the path length.
Given:
$A = 0.674$
$C = 1.58195 \times 10^{-4} \text{ mol/L}$
$L = 5 \text{ cm}$
Rearrange the formula to solve for $\epsilon$:

$$\epsilon = \frac{A}{C L}$$

Substitute the values:

$$\epsilon = \frac{0.674}{(1.58195 \times 10^{-4} \text{ mol/L}) \times (5 \text{ cm})}$$ $$\epsilon = \frac{0.674}{7.90975 \times 10^{-4} \text{ mol cm/L}}$$ $$\epsilon \approx 852.00 \text{ L mol}^{-1} \text{ cm}^{-1}$$

Rounding to two significant figures (due to $25 \text{ ppm}$), the molar absorptivity is:

$$\epsilon \approx 850 \text{ L mol}^{-1} \text{ cm}^{-1}$$

Molar absorptivity ($\epsilon$) is an intrinsic property of the substance at a specific wavelength. It does not depend on the cell path length. Therefore, the molar absorptivity remains the same regardless of the cell length.

The molar absorptivity of KMnO$_4$ at $515 \text{ nm}$ is $\boxed{\text{850 L mol}^{-1} \text{ cm}^{-1}}$.
The molar absorptivity in a $1 \text{ cm}$ cell will also be $\boxed{\text{850 L mol}^{-1} \text{ cm}^{-1}}$.