Question

A sample of gas occupies \(135 \mathrm{mL}\) at \(22.5^{\circ} \mathrm{C} ;\) the pressure is \(165 \mathrm{mm}\) Hg. What is the pressure of the gas sample when it is placed in a 252 -mL flask at a temperature of \(0.0^{\circ} \mathrm{C} ?\)

Short answer

The pressure of the gas in the 252 mL flask at 0.0°C is approximately 70.9 mm Hg.

Step-by-step solution

Understand the Problem

We are given the initial volume, temperature, and pressure of a gas sample. We are asked to find the new pressure after the volume and temperature change, while using the combined gas law that accounts for changes in pressure, volume, and temperature simultaneously.

Write Down the Combined Gas Law Equation

The combined gas law equation that relates initial and final states of a gas is given by: \[ \frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2} \]where \(P\) is the pressure, \(V\) is the volume, and \(T\) is the temperature in Kelvin.

Convert Temperatures from Celsius to Kelvin

To use the gas law, temperatures must be in Kelvin. Convert the initial temperature from Celsius to Kelvin by adding 273.15 to the Celsius temperature: \[ T_1 = 22.5 + 273.15 = 295.65 \, K \]Convert the final temperature in a similar way: \[ T_2 = 0.0 + 273.15 = 273.15 \, K \]

Plug Values into the Combined Gas Law Equation

Insert the known values into the equation: \[ \frac{165 \, \mathrm{mm} \, \mathrm{Hg} \times 135 \, \mathrm{mL}}{295.65 \, K} = \frac{P_2 \times 252 \, \mathrm{mL}}{273.15 \, K} \]This sets up the relationship to solve for \(P_2\).

Solve for the Unknown Pressure \(P_2\)

Rearrange the equation to isolate \(P_2\): \[ P_2 = \left(\frac{165 \, \mathrm{mm} \, \mathrm{Hg} \times 135 \, \mathrm{mL}}{295.65 \, K}\right) \times \left(\frac{273.15 \, K}{252 \, \mathrm{mL}}\right) \]Calculate \(P_2\) to find the new pressure:\[ P_2 \approx 70.9 \, \mathrm{mm} \, \mathrm{Hg} \]

Key concepts

Gas Pressure

Gas pressure is the force that gas molecules exert as they collide with the walls of their container. It is typically measured in units such as millimeters of mercury (mm Hg), atmospheres (atm), or pascals (Pa). When pressure increases, it indicates more frequent and forceful collisions, usually because the gas is compressed or heated.

• When volume decreases or temperature increases, pressure tends to rise if the amount of gas is constant.
• Conversely, if the volume increases or the temperature decreases, the pressure typically drops.

This relationship between pressure, volume, and temperature is fundamental in understanding gas behaviors and helps to predict changes using laws like the combined gas law.

Temperature Conversion

Temperature conversion is critical when working with gas laws, as they often require temperatures in Kelvin. Kelvin is the SI unit for temperature used in scientific calculations because it is an absolute temperature scale starting at absolute zero.

• The conversion from Celsius to Kelvin is straightforward: simply add 273.15 to the Celsius temperature. For example, converting 22.5°C results in 295.65 K, while 0°C converts to 273.15 K.
• This conversion ensures that all temperatures are positive and proportionally related to the kinetic energy of the gas molecules.

Correct temperature conversion is crucial for accurate calculations in the combined gas law and other gas equations.

Volume Change

Volume change occurs when you change the space in which gas molecules are contained. In the context of gas laws, the volume is usually expressed in liters or milliliters.

• Doubling the volume of a container can, under constant temperature, decrease the pressure by half, and vice versa, as long as the gas quantity remains unchanged.
• In the combined gas law, an initial volume is compared to a final volume, as in going from 135 mL to 252 mL, which affects the pressure or temperature changes.

Understanding how volume changes affect other properties of gases is key to predicting outcomes in experiments and natural phenomena involving gases.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that describes the relationship between pressure (P), volume (V), temperature (T), and the number of moles of gas (n). It is expressed as \( PV = nRT \), where \( R \) is the ideal gas constant.
Although not explicitly used in the combined gas law, the ideal gas law underpins its concepts because it assumes a perfect scenario where there is no interaction between gas particles and the particles occupy no volume.

• The ideal gas law helps in predicting the behavior of gases under different conditions and provides a baseline with which real gases can be compared.
• When working with the combined gas law, though it is typically about change in state, understanding the basis provided by the ideal gas law provides deeper insight into the nature of gases.

This foundational knowledge enables students to grasp more complex real-world gas behaviors.