A gas at constant temperature has an initial pressure of 1.5 atm and an initial volume of 200 ml. If the pressure is changed to 1.01 atm, what is the final volume?

Boyle's Law states that the product of the initial pressure and volume is equal to the product of the final pressure and volume:

$$P_1V_1 = P_2V_2$$

From the given steps, it appears you are trying to find the final volume, which is typically denoted as $V_2$. The equation shown in your work, $V^2 = \frac{P_1V_1}{P_2}$, should be $V_2 = \frac{P_1V_1}{P_2}$. We will proceed with this corrected formula.

$$V_2 = \frac{P_1V_1}{P_2}$$

Given:
$P_1 = 1.5 \text{ atm}$
$V_1 = 200 \text{ ml}$
$P_2 = 1.01 \text{ atm}$

Substitute these values:

$$V_2 = \frac{1.5 \text{ atm} \times 200 \text{ ml}}{1.01 \text{ atm}}$$

First, multiply the values in the numerator:

$$V_2 = \frac{300 \text{ atm} \cdot \text{ml}}{1.01 \text{ atm}}$$

Now, divide to find $V_2$:

$$V_2 = 297.0297... \text{ ml}$$

Rounding to two decimal places:

$$V_2 \approx 297.03 \text{ ml}$$

The final volume is:

$$\text{297.03 ml}$$