Question

A \(4.66 \mathrm{L}\) sample of gas is warmed from \(273 \mathrm{K}\) to a final temperature of \(398 \mathrm{K}\). Assuming no change in pressure, what is the final volume of the gas?

Short answer

Answer: The final volume of the gas is 6.8 L.

Step-by-step solution

Write down the given parameters

We are given the following parameters: Initial volume: \(V_1 = 4.66 L\) Initial temperature: \(T_1 = 273K\) Final temperature: \(T_2 = 398K\) We need to find the final volume: \(V_2\)

Apply Charles' Law

Charles' Law states that the volume of a gas at constant pressure is directly proportional to its temperature. Mathematically, this can be expressed as: \(\frac{V_1}{T_1} = \frac{V_2}{T_2}\) Now, we can plug in the given values and solve for \(V_2\).

Plug in the given values and solve for the final volume

Plugging the given values into the equation, we get: \(\frac{4.66 \mathrm{L}}{273 \mathrm{K}} = \frac{V_2}{398 \mathrm{K}}\) Now, to solve for \(V_2\), multiply both sides of the equation by \(398 \mathrm{K}\): \(V_2 = \frac{4.66 \mathrm{L}}{273 \mathrm{K}} \times 398 \mathrm{K}\)

Calculate the final volume

Perform the calculation: \(V_2 = \frac{4.66 \mathrm{L}}{273 \mathrm{K}} \times 398 \mathrm{K} = 6.8 \mathrm{L}\) So, the final volume of the gas is \(6.8 \mathrm{L}\).

Key concepts

Gas Laws

Gas laws are crucial in understanding how gases behave under different conditions. They govern the relationship between key variables such as temperature, volume, and pressure. There are several gas laws, but the most common among them include:

• Boyle's Law
• Charles' Law
• Avogadro's Law
• Gay-Lussac's Law

Each law describes how one specific pair of variables is related when the other factors remain constant. In the context of our exercise, Charles' Law is especially important. This law examines how the volume of a gas changes with its temperature when the pressure is held constant. It is a foundational concept within the study of gases and is often combined with others to form the ideal gas law, which is a comprehensive equation describing the state of a gas. Understanding these laws helps us predict and manipulate the behavior of gases in various practical and scientific applications.

Volume and Temperature Relationship

Charles' Law specifically deals with the volume and temperature relationship of a gas at constant pressure. It states that the volume of a gas is directly proportional to its temperature, as long as the pressure does not change. This can be put simply as: when you increase the temperature, the volume increases too.
Mathematically, Charles' Law is expressed as:

• \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \)

This formula lets us calculate what happens to the volume when there is a change in temperature. For example, if you heat a gas, it will expand if the pressure stays the same. The real-world applications of this are vast, from everyday experiences like inflating balloons to complex systems in engineering and physics. Being able to predict how a gas's volume will change with temperature change is essential for a wide range of scientific inquiries and industrial applications.

Ideal Gas Behavior

Ideal gas behavior refers to how gases are expected to act under certain theoretical conditions. An 'ideal gas' is one that follows the gas laws perfectly. In reality, no gas is truly ideal as every gas has its peculiarities and specific interactions. However, many gases behave approximately like an ideal gas under many standard conditions. For the purposes of calculations and predictions, gases are often assumed to be ideal. This simplification allows easy application of gas laws such as Charles' Law. Under conditions of low pressure and high temperature, real gases tend to act more like ideal gases because intermolecular forces become negligible. When gases deviate from ideal behavior, typically at very high pressures or very low temperatures, corrections need to be made for more accurate results. The Van der Waals equation, for example, accounts for these deviations by including factors for volume occupied by gas molecules and attractions between them. But, for our exercise, assuming ideal gas behavior makes it straightforward to use Charles’ Law and predict changes in the volume due to changes in temperature.