Master Boyle's Law Calculations | ChemistPath

To solve this problem, we can use Boyle's Law, which states that for a given mass of an ideal gas at a constant temperature, the volume of the gas is inversely proportional to its pressure.

1. Identify the Given Information:

Initial Volume (\(V_1\)) = \(250\,\text{mL}\)
Initial Pressure (\(P_1\)) = \(800\,\text{mmHg}\)
Final Volume (\(V_2\)) = \(500\,\text{mL}\)
Final Pressure (\(P_2\)) = ? (required in \(\text{atm}\))
Temperature (\(T\)) = Constant

2. Formula:

Boyle's Law is mathematically expressed as:

\[ P_1 V_1 = P_2 V_2 \]

Rearranging the formula to solve for the final pressure (\(P_2\)):

\[ P_2 = \frac{P_1 V_1}{V_2} \]

3. Calculation:

Step 1: Calculate the final pressure in \(\text{mmHg}\)
Substitute the given values into the rearranged equation:

\[ P_2 = \frac{800\,\text{mmHg} \times 250\,\text{mL}}{500\,\text{mL}} \] \[ P_2 = \frac{200{,}000\,\text{mmHg}\cdot\text{mL}}{500\,\text{mL}} \] \[ P_2 = 400\,\text{mmHg} \]

Step 2: Convert the pressure from \(\text{mmHg}\) to \(\text{atm}\)
We know the standard conversion factor between millimeters of mercury (\(\text{mmHg}\)) and atmospheres (\(\text{atm}\)) is:

\[ 1\,\text{atm} = 760\,\text{mmHg} \]

Now, divide the pressure in \(\text{mmHg}\) by \(760\):

\[ P_2 = \frac{400\,\text{mmHg}}{760\,\text{mmHg/atm}} \] \[ P_2 \approx 0.526\,\text{atm} \]

Answer:

The pressure needed to expand the gas sample to \(500\,\text{mL}\) at the same temperature is approximately \(0.526\,\text{atm}\).