Problem 145 - Chemistry A Molecular Approach

Question

The volume of a sample of a fixed amount of gas is decreased from \(2.0 \mathrm{~L}\) to \(1.0 \mathrm{~L}\). The temperature of the gas in kelvins is then doubled. What is the final pressure of the gas in terms of the initial pressure?

Step-by-Step Solution

Answer

The final pressure of the gas \(P_2\) is twice the initial pressure \(P_1\), so \(P_2 = 2P_1\).

Step 1: Understand the relationship between the variables involved

To determine the final pressure, comprehend the relationship between pressure (P), volume (V), and temperature (T) for a fixed amount of gas. This relationship is described by the combined gas law, \(\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}\), where the subscripts 1 and 2 refer to the initial and final states of the gas, respectively.

Step 2: Applying the Combined Gas Law

Set up the equation with the known quantities and variables. We know the initial volume \(V_1 = 2.0 \mathrm{L}\), the final volume \(V_2 = 1.0 \mathrm{L}\), and that the final temperature is twice the initial temperature \(T_2 = 2T_1\). We are looking for the final pressure \(P_2\) in terms of the initial pressure \(P_1\). Inputting these into the combined gas law equation gives \(\frac{P_1 \times 2.0}{T_1} = \frac{P_2 \times 1.0}{2T_1}\).

Step 3: Solve for the final pressure P2

Isolate \(P_2\) on one side of the equation to solve for the final pressure. Cross multiply and simplify to get \(P_2 = 2P_1\). Notice that \(T_1\) cancels out from both sides during this process.

Key Concepts

Gas Laws

Gas laws are fundamental principles in chemistry that describe the behavior of gases under different conditions of temperature, pressure, and volume. They are essential for understanding how gases respond to environmental changes and for predicting the outcome of experiments and industrial processes. The primary gas laws include Boyle's Law, which defines the inverse relationship between pressure and volume, and Charles's Law, which states that volume and temperature are directly related.

When combined, these principles lead to the combined gas law, which allows us to calculate one variable (pressure, volume, or temperature) if the other two are known. The law is particularly useful for situations where a gas undergoes simultaneous changes in pressure, volume, and temperature. This combined approach to understanding gas behavior helps students to anticipate how gases will react in different scenarios, furthering their grasp of chemical reactions and real-world applications in fields like meteorology and engineering.

Pressure-Volume-Temperature Relationship

The pressure-volume-temperature relationship is a cornerstone of the combined gas law. It presents a way to predict how a change in one variable will affect the others, provided the amount of gas remains constant. Understanding this relationship is crucial for solving problems related to gas behavior. For instance, as volume decreases, pressure, conversely, must increase if the temperature remains constant - this illustrates Boyle's Law. However, if the temperature rises, the volume tends to expand as well, exemplified by Charles's Law.

As we apply these principles to the combined gas law equation, \(\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}\), we can see how directly proportional relationships (volume and temperature) and inversely proportional relationships (pressure and volume) interact. This flexible framework allows students to analyze complex scenarios where multiple gas properties change simultaneously. It provides a predictive tool for understanding the outcomes in enclosed gases, like an air balloon or a cylinder in an engine, enhancing students' problem-solving skills in practical chemistry applications.

Solving Chemistry Problems

Solving chemistry problems often requires a methodical approach to dissect and comprehend the various components of a given question. Initially, one should identify the known variables and the unknowns that need to be calculated. For gas law problems, this includes recognizing the initial and final states for variables such as pressure (P), volume (V), and temperature (T).

Using formulas like the combined gas law and understanding the proportional relationships between variables enable students to construct equations based on the information given in a problem. The process of solving includes substituting the known values into the equation, simplifying, and solving for the unknown quantity. It is essential to keep units consistent and, where necessary, convert temperatures to Kelvin for accuracy. The ability to think critically and apply these steps in a systematic fashion is an invaluable skill for tackling a wide range of challenges in chemistry, bolstering students’ confidence and competence in the subject.