Question

Suppose a 125 -mL sample of argon is cooled from \(450 \mathrm{~K}\) to \(250 \mathrm{~K}\) at constant pressure. What will be the volume of the sample at the lower temperature?

Short answer

The volume of the 125-mL argon sample at the lower temperature of 250 K can be found using Charles's Law: \(\frac{V_1}{T_1} = \frac{V_2}{T_2}\). By substituting the given values (\(V_1 = 125\) mL, \(T_1 = 450\) K, and \(T_2 = 250\) K), we can calculate the final volume, \(V_2\), which is approximately 69.44 mL.

Step-by-step solution

Write down the given information

We are given the following information: - Initial volume, \(V_1 = 125\) mL - Initial temperature, \(T_1 = 450\) K - Final temperature, \(T_2 = 250\) K - Pressure, constant Our goal is to find the final volume, \(V_2\).

Apply Charles's Law formula

We can now apply Charles's Law with the given information: \[\frac{V_1}{T_1} = \frac{V_2}{T_2}\]

Substitute the values

We can substitute the given values in the formula: \[\frac{125\text{ mL}}{450\text{ K}} = \frac{V_2}{250\text{ K}}\]

Solve for the final volume, \(V_2\)

To find the value of \(V_2\), we can cross-multiply and solve for \(V_2\): \[V_2 = \frac{125\text{ mL} \cdot 250\text{ K}}{450\text{ K}}\]

Calculate the final volume

Now, let's calculate the final volume, \(V_2\), upon simplification: \[V_2 = \frac{31250\text{ mL\cdot.K}}{450\text{ K}} \approx 69.44\text{ mL}\] So the volume of the argon sample at the lower temperature of 250 K would be approximately 69.44 mL.

Key concepts

Gas Laws

In the realm of chemistry, gas laws like Charles's Law are essential in understanding gas behavior under varying conditions. Gas laws are mathematical relationships that describe properties of gases such as pressure, volume, and temperature. Each law provides specific rules on how these variables interact.

Charles's Law focuses on the relationship between volume and temperature. It states that for a fixed mass of gas at constant pressure, the volume of the gas is directly proportional to its temperature in Kelvin. This means as the temperature increases, the volume expands, and when the temperature decreases, the volume shrinks. Understanding these laws helps predict how gases will react to changes in their environment, which is crucial for many scientific calculations and real-world applications.

It's important to remember that these laws apply ideally. Real gases approximate these behaviors under certain conditions, typically low pressure and high temperature, where interactions between gas particles are minimal.

Thermodynamics

Thermodynamics is the branch of physical science that deals with the relations between heat and other forms of energy. Within thermodynamics, various principles describe how energy is transferred in physical and chemical processes. It connects to gas laws as it explains how temperature, a measure of thermal energy, affects gas properties.

In the context of the exercise with argon gas, thermodynamics provides the framework for understanding heat transfer's impact on gas volume. When the argon is cooled, heat energy is removed, resulting in a decrease in temperature. According to Charles's Law, this temperature change causes a decrease in volume when pressure is constant.

Moreover, thermodynamic principles help in correlating these observed changes in gas behavior with theoretical models. This quantitative analysis allows scientists and engineers to predict how gases will behave under varied thermal conditions.

Temperature and Volume Relationship

The relationship between temperature and volume in gases is a fundamental concept in the study of gas behaviors. Charles's Law explicates this relationship, stating that at constant pressure, the volume of a gas is directly proportional to its temperature in Kelvin.

To see this in action, consider the given example. As the argon's temperature drops from 450 K to 250 K, its volume also decreases. Mathematically, this relationship is represented as \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \). This formula demonstrates that changes in temperature result in proportional changes in volume.

Such understanding is vital for practical applications like designing engines, refrigeration systems, and even predicting weather patterns, all of which rely on the predictable nature of temperature and volume interactions in gases. By monitoring these changes, it becomes possible to control and utilize gases effectively in various industries.