The initial volume of a gas cylinder is 750.0 mL. If the pressure of gas inside the cylinder changes from 840.0 mmHg to 360.0 mmHg, the final volume the gas will be:

According to Boyle's Law, for a fixed amount of gas at constant temperature, the product of pressure and volume remains constant. This means that the initial pressure multiplied by the initial volume equals the final pressure multiplied by the final volume. So, we can write the equation as $$ P_1 V_1 = P_2 V_2 $$.

In this problem, the initial volume $$ V_1 $$ is given as 750.0 mL, the initial pressure $$ P_1 $$ is 840.0 mmHg, and the final pressure $$ P_2 $$ is 360.0 mmHg. We need to find the final volume $$ V_2 $$. Rearranging Boyle's Law to solve for $$ V_2 $$, we get $$ V_2 = \frac{P_1 V_1}{P_2} $$.

Now, substitute the given values into the equation: $$ V_2 = \frac{840.0 \text{mmHg} \times 750.0 \text{mL}}{360.0 \text{mmHg}} $$. Notice that the units of mmHg will cancel out, leaving the volume in mL.

First, calculate the numerator: $$ 840.0 \times 750.0 $$. Multiply 840 by 750. Break it down: 840 multiplied by 700 is 588,000, and 840 multiplied by 50 is 42,000. Adding these together gives 588,000 + 42,000 = 630,000. Since both 840.0 and 750.0 have one decimal place, the product is 630,000.00, but we can work with 630,000 for simplicity.

Now, divide this result by the final pressure: $$ V_2 = \frac{630,000}{360.0} $$. To make the division easier, we can eliminate the decimal in the denominator by multiplying both numerator and denominator by 10: $$ V_2 = \frac{630,000 \times 10}{360.0 \times 10} = \frac{6,300,000}{3600} $$.

Simplify this fraction by dividing both numerator and denominator by 100: $$ \frac{6,300,000}{3600} = \frac{63,000}{36} $$. Now, divide 63,000 by 36. Calculate step by step: 36 multiplied by 1,000 is 36,000. Subtract 36,000 from 63,000 to get 27,000. Now, 36 multiplied by 750 is 27,000 (since 36 × 700 = 25,200 and 36 × 50 = 1,800, and 25,200 + 1,800 = 27,000). Adding the parts: 1,000 + 750 = 1,750. So, $$ \frac{63,000}{36} = 1,750 $$.

Therefore, the final volume is 1,750 mL. However, the options are given in liters (L). Convert mL to L by dividing by 1,000 since 1 L = 1,000 mL. So, $$ V_2 = \frac{1,750 \text{mL}}{1,000} = 1.750 \text{L} $$.

Hence, the correct answer is Option A.