Question

A gas has an initial pressure of 1.445 atm and an initial volume of \(1.009 \mathrm{~L}\). What is its new pressure if volume is changed to \(0.556 \mathrm{~L}\) ? Assume temperature and amount are held constant.

Short answer

The new pressure is approximately 2.620 atm.

Step-by-step solution

Identify the Known Variables

We are given the initial pressure \( P_1 = 1.445 \text{ atm} \) and initial volume \( V_1 = 1.009 \text{ L} \). The final volume \( V_2 \) is given as \( 0.556 \text{ L} \). We need to find the final pressure \( P_2 \).

Recall the Relationship Used

Since the temperature and amount of gas are constant, we use Boyle's Law which states that the pressure of a gas is inversely proportional to its volume. This law is mathematically stated as \( P_1 V_1 = P_2 V_2 \).

Rearrange Boyle's Law Equation

Rearrange the equation \( P_1 V_1 = P_2 V_2 \) to solve for the final pressure \( P_2 \). This gives us \( P_2 = \frac{P_1 V_1}{V_2} \).

Substitute Known Values into the Equation

Substitute the given values from Step 1 into the rearranged Boyle's law equation: \( P_2 = \frac{1.445 \times 1.009}{0.556} \).

Calculate the Final Pressure

Perform the multiplication and the division. First multiply \( 1.445 \times 1.009 = 1.457005 \). Then divide by the final volume: \( \frac{1.457005}{0.556} \approx 2.620 \text{ atm} \).

Key concepts

Pressure-Volume Relationship

The pressure-volume relationship in gases is a fundamental concept in understanding how gases behave. According to Boyle's Law, if the temperature and the amount of gas are kept constant, the pressure and volume of the gas are inversely related. This means that as one increases, the other decreases. This relationship can be written mathematically as \( P_1V_1 = P_2V_2 \), where \( P_1 \) and \( V_1 \) are the initial pressure and volume, and \( P_2 \) and \( V_2 \) are the final pressure and volume respectively.
When the volume of a gas decreases, the particles are compressed into a smaller space, leading to more frequent collisions with the walls of the container. This increase in collisions results in a higher pressure. Conversely, when the volume increases, the pressure decreases because the gas particles have more space to move freely.

Gas Laws

Gas laws are a set of rules that describe how different properties of a gas—such as pressure, volume, and temperature—are related. These laws help us predict how a gas will react when conditions change. Boyle's Law is one component of these gas laws, specifically dealing with pressure and volume changes.
Other important gas laws include:

• Charles’s Law: Deals with the relationship between volume and temperature, indicating that volume increases as temperature increases, when pressure is constant.
• Avogadro’s Law: States that volume is directly proportional to the number of gas moles, given constant temperature and pressure.
• Ideal Gas Law: It combines these simpler laws into a single equation \( PV = nRT \), where \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin.

Each of these laws applies under specific conditions and provides a useful tool for solving gas-related problems.

Inverse Proportionality

Inverse proportionality is a key concept in understanding Boyle's Law. It occurs when two variables are related in such a way that if one variable increases, the other decreases. Mathematically, this relationship is represented as \( y \propto \frac{1}{x} \), or in the context of Boyle's Law, \( P \propto \frac{1}{V} \).

In a real-world scenario, imagine squeezing a balloon. As the balloon's volume decreases because you press it down, the air particles inside have less space to move, causing an increase in pressure. This simple action elegantly demonstrates inverse proportionality and how it applies to gases.
Understanding inverse proportionality is crucial in many areas of science and engineering, as it provides insights into how different systems respond to changes in conditions.