Question

The volume of a gas is \(7.15 \mathrm{~L},\) measured at \(1.00 \mathrm{~atm} .\) What is the pressure of the gas in \(\mathrm{mmHg}\) if the volume is changed to \(9.25 \mathrm{~L}\) ? (The temperature remains constant.)

Short answer

The pressure of the gas is approximately 587.86 mmHg.

Step-by-step solution

Identify Given Information

We are provided with the initial conditions of the gas: the initial volume \( V_1 = 7.15 \) L and the initial pressure \( P_1 = 1.00 \) atm. The final volume \( V_2 \) is given as \( 9.25 \) L. We aim to find the final pressure \( P_2 \) in mmHg.

Apply Boyle's Law

Since the temperature remains constant, we can use Boyle's Law which states that for a constant temperature, the product of pressure and volume is constant. This is expressed by the formula \( P_1 V_1 = P_2 V_2 \).

Solve for Final Pressure

Rearrange the formula from Boyle's Law to solve for \( P_2 \): \( P_2 = \frac{P_1 \, V_1}{V_2} \). Substitute the known values: \( P_2 = \frac{1.00 \text{ atm} \times 7.15 \text{ L}}{9.25 \text{ L}} \).

Perform Calculation

Calculate the value: \( P_2 = \frac{7.15}{9.25} \approx 0.7735 \text{ atm} \).

Convert Pressure to mmHg

Convert the pressure from atmospheres to mmHg using the conversion factor \(1 \text{ atm} = 760 \text{ mmHg}\). Thus, \( P_2 = 0.7735 \text{ atm} \times 760 \text{ mmHg/atm} \approx 587.86 \text{ mmHg} \).

Key concepts

Gas Laws

Gas laws are a collection of laws that describe the behavior of gases. They relate the pressure, volume, temperature, and amount of gas. Among them is Boyle's Law, which specifically addresses the relationship between pressure and volume, assuming the temperature and the number of gas particles remain constant.

These laws are derived from the ideal gas law equation: \[ PV = nRT \] where

• \(P\) is the pressure of the gas,
• \(V\) is the volume of the gas,
• \(n\) is the number of moles,
• \(R\) is the ideal gas constant, and
• \(T\) is the temperature in Kelvin.

Understanding these laws helps predict the behavior of gases under different conditions, which is particularly useful in fields ranging from chemistry to meteorology.

Pressure-Volume Relationship

The pressure-volume relationship, or Boyle's Law, is a fundamental gas law that describes how the pressure of a gas tends to decrease as the volume of the gas increases, provided the temperature remains constant. This is expressed mathematically as: \[ P_1 V_1 = P_2 V_2 \] where

• \(P_1\) and \(P_2\) refer to the initial and final pressures, respectively, and
• \(V_1\) and \(V_2\) refer to the initial and final volumes.

When you pressurize a gas, its molecules are forced into a smaller volume, leading to an increase in pressure. Conversely, increasing the space for gas molecules allows the pressure to drop. Boyle's Law is an essential part of understanding how gases behave under pressure changes.

Pressure Conversion

Pressure conversion is crucial when working with gas laws, as pressure can be measured in various units, such as atmospheres (atm), millimeters of mercury (mmHg), or Pascals (Pa). To ensure calculations are correct, it is often necessary to convert pressure from one unit to another.

For example, the conversion factor between atmospheres and millimeters of mercury is based on the fact that:

• 1 atm = 760 mmHg

In our problem, we calculated the final pressure as approximately 0.7735 atm. To convert this to mmHg, we multiply by 760: \[ P_2 = 0.7735 ext{ atm} \times 760 ext{ mmHg/atm} = 587.86 ext{ mmHg} \] Being familiar with these conversions allows one to navigate between different systems of measurement accurately.

Constant Temperature Process

A constant temperature process, often occurring in applied gas problems, is also known as an isothermal process. During such a process, the temperature of the gas remains constant, which significantly simplifies calculations involving gases.

In the context of the pressure-volume relationship, maintaining a constant temperature means that any change in the volume will directly affect the pressure and vice versa. Since the temperature does not change, the energy distribution among gas molecules remains steady, and only their spatial distribution changes.

When employing Boyle's Law in calculations, the assumption of a constant temperature allows us to focus exclusively on solving for unknown pressures or volumes, confidently knowing temperature does not influence the outcome.