Problem 32 - Chemistry A Molecular Approach

Question

A sample of gas has an initial volume of \(13.9 \mathrm{~L}\) at a pressure of 1.22 atm. If the sample is compressed to a volume of \(10.3 \mathrm{~L}\) what is its pressure?

Step-by-Step Solution

Answer

The final pressure of the gas is 1.60 atm after being compressed to a volume of 10.3 L.

Step 1: Identify the Known Values

From the problem, we know the initial volume (V1) is 13.9 L, the initial pressure (P1) is 1.22 atm, and the final volume (V2) is 10.3 L. We are asked to find the final pressure (P2).

Step 2: Write Down the Formula for Boyle's Law

Boyle's Law states that for a fixed amount of an ideal gas kept at a fixed temperature, pressure and volume are inversely proportional. The formula is: P1 * V1 = P2 * V2.

Step 3: Rearrange the Equation to Solve for P2

We can rearrange the formula to solve for P2 by dividing both sides of the equation by V2. This gives us: P2 = (P1 * V1) / V2.

Step 4: Plug in the Known Values

Substitute the known values into the rearranged equation: P2 = (1.22 atm * 13.9 L) / 10.3 L.

Step 5: Calculate P2

Perform the calculation to find P2: P2 = (1.22 atm * 13.9 L) / 10.3 L = 16.518 atm / 10.3 L = 1.60 atm.

Step 6: Round to Appropriate Significant Figures

The given pressures are to two decimal places, so we should round our answer to P2 to two decimal places as well: P2 = 1.60 atm.

Key Concepts

Gas Laws

Understanding gas laws is crucial to grasp how gases react to changes in pressure, temperature, and volume. Boyle's Law is one of the fundamental gas laws that explain the pressure-volume relationship of a gas at constant temperature. In addition to Boyle's Law, there are others like Charles's Law, which deals with temperature and volume, and Avogadro's Law, relating volume and the number of gas particles. These laws lay the foundation for the ideal gas law which combines all the variables. In real-world scenarios, these concepts help us predict how a gas will behave under different conditions, such as inflating a balloon or using a syringe.

For example, divers need to understand gas laws to avoid complications like decompression sickness, which is related to changes in pressure. Moreover, engineers use these laws to design various devices and systems that operate under a range of gaseous conditions. Ensuring a firm understanding of these principles can be an essential asset across numerous fields including meteorology, cooking, and even astrophysics.

Pressure-Volume Relationship

Boyle's Law shows the inverse relationship between the pressure and volume of a gas. According to this law, if the temperature remains constant, increasing the volume of the gas will decrease its pressure, and vice versa. This is called an inverse relationship because the product of pressure and volume is a constant for a given amount of gas at a fixed temperature.

For instance, in our exercise, when the volume of the gas was decreased from 13.9 liters to 10.3 liters, the pressure naturally increased. This is precisely because the gas molecules have less space to move around, and hence they collide with the walls of the container more frequently, leading to a rise in pressure. This principle has numerous applications including in the operation of piston engines, where the combustion of fuel increases the pressure in the chamber, forcing the piston to move and create mechanical energy.

Ideal Gas Behavior

An ideal gas is a hypothetical gas whose particles occupy negligible space and have no interactions except when they collide elastically. This concept simplifies the study of gas behavior because it allows the use of straightforward mathematical relationships like Boyle's Law to predict how a gas will respond to changes in pressure, volume, and temperature. These relationships assume the gas behaves ideally, which is a good approximation for many gases under standard conditions.

However, at high pressures or low temperatures, real gases deviate from ideal behavior. Despite these limitations, the ideal gas model is essential in chemical engineering and environmental science to understand and model the behavior of gases in various processes and conditions. For example, calculations of the volume of reactant gases in a chemical reaction or predicting the behavior of the atmosphere all rely on the principles of ideal gas behavior.