Question

Suppose a 375-mL sample of neon gas at 78 ' \(C\) is cooled to 22 ' \(C\) at constant pressure. What will be the new volume of the neon sample?

Short answer

The new volume of the neon sample when cooled to 22 'C at constant pressure will be approximately \(315 \text{ mL}\).

Step-by-step solution

Convert temperatures to Kelvin

We need to add 273.15 to both initial and final temperatures. Initial temperature: \(T_1 = 78 + 273.15 = 351.15 \;K\) Final temperature: \(T_2 = 22 + 273.15 = 295.15 \;K\)

Set up the equation using Charles's Law

Charles's Law states that if the pressure of a gas remains constant, the ratio of the initial volume to the initial temperature will be equal to the ratio of the final volume to the final temperature: \(V_1/T_1 = V_2/T_2\) Plug in the known values: \(\frac{375 \text{ mL}}{351.15 \text{ K}} = \frac{V_2}{295.15 \text{ K}}\)

Solve for the new volume (\(V_2\))

To solve for \(V_2\), we can cross-multiply and then divide by the final temperature: \(V_2 = \frac{375 \text{ mL} \times 295.15 \text{ K}}{351.15 \text{ K}}\) Now, we perform the calculation: \(V_2 = \frac{110643.75 \text{ mL·K}}{351.15 \text{ K}} = 314.98 \text{ mL}\) The new volume of the neon sample when cooled to 22 'C at constant pressure will be approximately \(315 \text{ mL}\).

Key concepts

Gas Laws in Chemistry

Understanding gas laws in chemistry is crucial for explaining how gases behave under various conditions of temperature, volume, and pressure. These laws provide a foundational concept for not only chemistry students but also for those pursuing physics and engineering.

One of the fundamental statements within this field is that gases consist of particles in constant, random motion, and their behavior can be described through simple mathematical relationships. A primary example of such a relationship is the one described by Charles's Law, which is strictly for scenarios where the pressure is constant.

Other vital gas laws include Boyle's Law, which examines pressure-volume at constant temperature; Gay-Lussac's Law, which involves pressure-temperature at constant volume; and Avogadro's Law, stating that volume is proportional to the amount of gas at a constant temperature and pressure. These laws are often combined into the combined gas law or further generalized into the ideal gas law, which applies when a gas behaves 'ideally', or close to a perfect theoretical model.

Visualizing Gas Particle Motion

A helpful way to grasp these concepts is by imagining a fixed amount of gas particles in a container. If you heat the gas, the particles move faster, taking up more space, and if contained, increasing the pressure. Conversely, cooling the gas slows the particles down, causing them to take up less space, thus decreasing volume when pressure is constant.

Gas laws are pivotal in real-world applications, from the way we understand atmospheric phenomena to the workings of engines and refrigeration systems.

Temperature-Volume Relationship

The temperature-volume relationship for gases, often known as Charles's Law, is a cornerstone of thermodynamic studies. It tells us that for a fixed amount of gas at constant pressure, the volume is directly proportional to its temperature measured in Kelvin.

Why Kelvin, you might ask? This unit of measurement begins at absolute zero, which is the theoretical temperature where particles have minimum thermal motion, which makes the relationships in gas behaviors more proportionate and calculations more precise.

When you encounter a problem involving a gas where the temperature changes, you can expect the volume to change as well. If the temperature increases, the volume will do the same to accommodate the faster-moving particles. Conversely, a drop in temperature will lead to a reduction in volume.

Real-Life Implications

Understanding this relationship has practical implications. For example, when inflating a balloon on a warm day, it expands as the air inside increases in volume with the rising temperature. If the balloon is then taken to a cooler environment, it shrinks as the air inside decreases in volume. The equation \(V_1/T_1 = V_2/T_2\) captures this precise relationship, allowing for predictions and calculations of volume changes following temperature shifts.

Ideal Gas Law Applications

The ideal gas law, represented by the equation \(PV = nRT\), where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is a constant, and \(T\) is the temperature, combines the principles of Boyle’s, Charles’s, and Avogadro’s laws. It serves as a critical tool for predicting and understanding the behavior of gases under a variety of conditions.

Applications of the ideal gas law are extensive in industries and research fields. In meteorology, it helps determine the density and pressure of the atmosphere, which are critical for weather prediction models. Engineers use it in designing aerodynamic vehicles and aircraft where pressure and temperature vary significantly.

In chemistry, it is foundational in stoichiometry calculations involving gaseous reactions. It's also used to determine molar masses, to calculate the amount of reactants and products in a chemical reaction, and to understand the behavior of gases during changes in state, such as evaporation and condensation.

Practical Considerations

While the law assumes ideal conditions — non-interacting gas particles that occupy no volume — in reality, gases exhibit ideal behavior only under low pressure and high temperature. Therefore, corrections with the van der Waals equation might be necessary for high pressure and low temperature scenarios to get accurate predictions. Equipped with the knowledge of the ideal gas law, students can delve into real-world problem-solving, predicting how gases will react to changes in their environment.