Question

The volume of a sample of ideal gas is proportional to its temperature \((\mathrm{K})\) at constant pressure.

Short answer

Under constant pressure, the volume of an ideal gas is directly proportional to its temperature, as described by Charles' Law (\(V \propto T\)). This relationship can be represented as \(\frac{V_{1}}{T_{1}} = \frac{V_{2}}{T_{2}}\). By knowing the initial volume and temperature, along with either the final volume or final temperature, you can solve for the unknown variable, demonstrating the proportional relationship between volume and temperature.

Step-by-step solution

Understanding Charles' Law

Charles' Law states that the volume (V) of an ideal gas is directly proportional to its temperature (T) when the pressure (P) and the amount of gas (n) are constant. In mathematical terms, this can be represented as: \(V \propto T\) To remove the proportionality sign, we introduce a constant (k): \(V = kT\) Here, k is the constant of proportionality.

Constant Pressure

In this exercise, we are given that the pressure of the ideal gas is constant. This means that the effect of pressure on the gas is not changing, so we only need to focus on the relationship between the volume (V) and temperature (T).

Volume and Temperature Relationship

Using the equation derived in Step 1, we can determine the relationship between the volume and temperature of the gas. Since the pressure is constant and we are dealing with an ideal gas, we can relate the initial volume-temperature ratio to the final volume-temperature ratio: \(\frac{V_{1}}{T_{1}} = \frac{V_{2}}{T_{2}}\)

Solving for Unknown Variables

If you are given enough information to determine the initial volume (V1) and temperature (T1), as well as either the final volume (V2) or the final temperature (T2), the above equation can be used to solve for the unknown variable. For example, if you are given: - Initial volume (V1) = 1 L - Initial temperature (T1) = 300 K - Final volume (V2) = 1.5 L You can now use the equation to solve for the final temperature (T2): \(\frac{1}{300} = \frac{1.5}{T_{2}}\) Multiply both sides by \(300T_{2}\): \(T_{2} = \frac{1.5 * 300}{1}\) \(T_{2} = 450\, K\) So, under constant pressure, when the volume of the gas increases from 1 L to 1.5 L, the temperature increases from 300 K to 450 K, which supports Charles' Law.

Key concepts

Ideal Gas

An ideal gas is a hypothetical gas that perfectly fits all the assumptions of the kinetic molecular theory. In an ideal gas, particles do not attract or repel each other, and their only interactions occur through elastic collisions. The particles are assumed to have negligible volume. This simplification allows us to predict and calculate the behavior of gases using mathematical models.

In reality, no gas perfectly follows the ideal gas law, but many gases behave approximately like an ideal gas under a wide range of conditions, particularly at high temperatures and low pressures. The ideal gas law serves as a useful approximation to understand the behavior of gases in these conditions.

Proportionality Constant

In the context of Charles' Law, the proportionality constant (k) is a value that relates the volume of a gas to its temperature under constant pressure. This constant is crucial because it allows the transformation of a proportional relationship into an equation.

The equation for Charles' Law is written as:

• \( V = kT \)

Understanding the proportionality constant helps us move from the qualitative to the quantitative. With the introduction of this constant, we can predict exactly how much the volume will change with a change in temperature, as long as the pressure doesn’t vary.

Temperature-Volume Relationship

Charles' Law presents the temperature-volume relationship of gases, emphasizing how the volume of a gas changes with temperature at a constant pressure. If you increase the temperature of a gas, its volume increases, and vice versa, assuming pressure remains unchanged.

Mathematically, this principle is shown as:

• \( \frac{V_{1}}{T_{1}} = \frac{V_{2}}{T_{2}} \)

This formula shows the direct proportional relationship between temperature and volume. When temperature is higher, gas molecules move faster, causing them to spread out and take up more volume. Conversely, a decrease in temperature will slow down the molecules, reducing the volume.

Gas Laws

Gas laws encompass a set of rules that describe the behavior of gases. Charles' Law, along with other laws like Boyle’s Law and Avogadro’s Law, form the foundation of the ideal gas law.

Key gas laws include:

• **Boyle’s Law:** At constant temperature, the volume of a gas is inversely proportional to its pressure (\( PV = ext{constant} \)).
• **Avogadro’s Law:** At constant temperature and pressure, the volume of a gas is directly proportional to the number of moles (\( V ig/ n = ext{constant} \)).

Together, these laws are integrated into the ideal gas equation:

• \( PV = nRT \)

where \( R \) is the ideal gas constant. Understanding these interrelated gas laws allows for the prediction and calculation of how gases will behave under varying conditions of temperature, volume, and pressure.