Question

What volume of \(\mathrm{H}_{2}\) must be collected at 725 torr to have \(0.550 \mathrm{~L}\) of the gas at \(1.00 \mathrm{~atm}\) ?

Short answer

The volume of the \(\mathrm{H}_{2}\) gas at 725 torr must be \( \frac{760 \, \mathrm{torr} \times 0.550 \, \mathrm{L}}{725 \, \mathrm{torr}} \) L.

Step-by-step solution

Identify the initial and final conditions

The initial volume \(V1\) of the \(\mathrm{H}_{2}\) gas is given as 0.550 L at the initial pressure \(P1\) of 1.00 atm. The final pressure \(P2\) is given as 725 torr. We need to find the final volume \(V2\).

Convert all pressures to the same units

All pressures should be expressed in the same unit. We know that 1 atm is equal to 760 torr. So, converted, the initial pressure \(P1\) is \(1.00 \, \mathrm{atm} \times \frac{760 \, \mathrm{torr}}{1 \, \mathrm{atm}} = 760 \, \mathrm{torr} \). The final pressure \(P2\) is given as 725 torr.

Apply Boyle's law

Since the temperature is constant, we can use Boyle's law, \(P1 \times V1 = P2 \times V2\). Plugging in the known values we get \(760 \, \mathrm{torr} \times 0.550 \, \mathrm{L} = 725 \, \mathrm{torr} \times V2\).

Calculate the unknown volume \(V2\)

To solve for \(V2\), divide both sides by 725 torr. Thus, \(V2 = \frac{760 \, \mathrm{torr} \times 0.550 \, \mathrm{L}}{725 \, \mathrm{torr}} \)

Key concepts

Gas Laws

Gas laws describe how gases behave under different conditions of pressure, volume, and temperature. An essential principle among these is Boyle's Law, which specifically addresses the relationship between pressure and volume of a gas when the temperature remains constant.
Boyle's Law states that for a fixed amount of gas at constant temperature, the volume of the gas is inversely proportional to its pressure. This means that when the pressure increases, the volume decreases, and vice versa.

• If you double the pressure on a gas, its volume is halved.
• Mathematically, this relationship can be represented as: \( P_1 \times V_1 = P_2 \times V_2 \). Here, \( P \) stands for pressure and \( V \) stands for volume.

Understanding this law helps us predict how a gas will compress or expand when the pressure changes. By knowing two conditions of pressure and volume, you can calculate the third, given the temperature stays constant.

Pressure Conversion

Pressure conversion is crucial in gas law calculations, as gas pressures may be given in different units. Common units of pressure include atmospheres (atm) and torr. To make accurate calculations, you must convert all pressures to equivalent units.
One atmosphere is equivalent to 760 torr. This conversion factor allows you to easily switch between these units by straightforward multiplication or division.

• To convert from atmospheric pressure to torr, multiply by 760. For example, \(1 \, \text{atm} \times 760 \, \frac{\text{torr}}{\text{atm}} = 760 \, \text{torr}\).
• To convert from torr to atmospheres, divide by 760. For example, \(725 \, \text{torr} \div 760 \, \frac{\text{torr}}{\text{atm}} \approx 0.953 \, \text{atm}\).

Conversion accuracy can impact your calculations. Hence, always verify that your pressure values are in the appropriate units before applying mathematical formulas like Boyle's Law.

Volume Calculation

Volume calculation is often needed to determine what happens to a gas when its conditions change. Using Boyle's Law, you can calculate the change in volume when pressure changes, assuming temperature is constant.First, ensure that all pressures are in the same unit. Then apply the formula from Boyle's Law:
\[ P_1 \times V_1 = P_2 \times V_2 \]In this formula, \( V_1 \) is the initial volume, \( P_1 \) is the initial pressure, \( P_2 \) is the final pressure, and \( V_2 \) is the final volume. To solve for \( V_2 \), rearrange the equation:

• Divide both sides by the final pressure \( P_2 \):
• \( V_2 = \frac{P_1 \times V_1}{P_2} \)
• Substitute the known values to find \( V_2 \).

This method lets you find the new volume of the gas after a change in pressure, making it a vital step in many gas law calculations.