Question

A 5.0 -m \(L\), sample of \(\mathrm{CO}_{2}\) gas is enclosed in a gas-tight syringe (see Figure 12.4 ) at \(22^{\circ}\) C. If the syringe is immersed in an ice bath \(\left(0^{\circ} \mathrm{C}\right),\) what is the new gas volume, assuming that the pressure is held constant?

Short answer

The new gas volume is approximately 4.63 mL.

Step-by-step solution

Analyze the Situation

We are asked to find the new volume of a gas when its temperature is decreased, with pressure held constant. This is a classic application of Charles's Law. Charles's Law states that \( V_1/T_1 = V_2/T_2 \) when pressure is constant, where \( V \) is the volume, and \( T \) is the temperature in Kelvin.

Convert Celsius to Kelvin

Temperatures must be converted from Celsius to Kelvin to apply Charles's Law. To convert, add 273.15 to the Celsius temperature. So, the initial temperature \( T_1 = 22^{\circ}C + 273.15 = 295.15 \ K \) and the final temperature \( T_2 = 0^{\circ}C + 273.15 = 273.15 \ K \).

Apply Charles's Law Formula

Use the formula \( V_1/T_1 = V_2/T_2 \). We know \( V_1 = 5.0 \ mL \). Substitute the known values: \( \frac{5.0}{295.15} = \frac{V_2}{273.15} \).

Solve for Unknown Volume \( V_2 \)

Cross-multiply to solve for \( V_2 \): \( V_2 = \frac{5.0 \ mL \times 273.15 \ K}{295.15 \ K} \). Calculate \( V_2 \): \( V_2 \approx 4.63 \ mL \).

Key concepts

Gas Laws

Gas laws are fundamental principles that describe how gases behave under various conditions. They provide relationships between variables like pressure, volume, and temperature. These laws were established through extensive observation and experimentation. The most well-known gas laws include Boyle's Law, Charles's Law, and Avogadro's Law. Each law isolates two of these variables to show how they relate when the rest remain constant.

• Boyle's Law states that the pressure of a gas is inversely proportional to its volume, when temperature is held constant.
• Charles's Law shows a direct proportionality between the volume and temperature of a gas, with pressure remaining constant.
• Avogadro's Law indicates that the volume of a gas is directly proportional to the number of moles of gas, at constant temperature and pressure.

Understanding these laws is crucial for predicting the behavior of gases in different scenarios. They form the foundation for the ideal gas law, which combines all these principles into one comprehensive equation.

Volume-Temperature Relationship

The volume-temperature relationship of gases is captured by Charles's Law. This law provides a simple yet powerful way to understand how gases expand or contract with temperature changes, as long as pressure stays constant. In any gas, molecules move more energetically as temperature increases, causing them to occupy more space. Conversely, when a gas cools, its molecular movement slows, leading to a reduction in volume.
For example, as detailed in the original exercise, we have a sample of gas with an initial volume and temperature. When the temperature decreases from 22°C to 0°C, Charles's Law predicts how the volume will also decrease based on the relationship:\[\frac{V_1}{T_1} = \frac{V_2}{T_2}\] This formula allows us to compute the final volume, making it a practical tool in experiments and real-world applications where conditions vary.

Ideal Gas Behavior

In science, understanding "ideal gas behavior" is key to predicting how gases will act under various conditions. An ideal gas is a theoretical concept in which the gas follows all the gas laws precisely without any deviations. Most gases at low pressures and high temperatures approximate this behavior, adhering closely to laws like Charles's Law.

• Ideal gases assume that molecules are point particles with no volume of their own.
• There are no intermolecular forces acting between the particles in an ideal gas.
• Molecular collisions are perfectly elastic, meaning no energy is lost in these interactions.

While no real gas fits this model perfectly, the ideal gas law provides a valuable framework for understanding gas behavior. It allows scientists to predict how gases will respond to changes in temperature, pressure, and volume. By making use of this behavior, adjustments and calculations, such as those made in the syringe experiment, are simplified.