Question

If the pressure on a \(2.10 \times 10^{4} \mathrm{~mL}\) sample of gas is doubled at constant temperature, what will be the new volume of the gas?

Short answer

The new volume of the gas when the pressure is doubled at constant temperature is \(1.05 \times 10^{4} \mathrm{~mL}\).

Step-by-step solution

Identify the given information

We are given the initial volume (\(V_1\)) of the gas, which is \(2.10 \times 10^{4} \mathrm{~mL}\), and we know that the pressure is being doubled (so, \(P_2 = 2P_1\)). We need to find the new volume of the gas (\(V_2\)).

Set up the equation using Boyle's Law

According to Boyle's Law, we have the equation: \(P_1V_1 = P_2V_2\) Since we know that the final pressure, \(P_2\), is twice the initial pressure, \(P_1\), we can replace \(P_2\) with \(2P_1\): \(P_1V_1 = (2P_1)V_2\)

Solve for the new volume, \(V_2\)

Next, we need to solve for \(V_2\). Since the question asks for the new volume of the gas, the final pressure (\(P_2\)) and initial pressure (\(P_1\)) are not relevant for our calculation. We can simplify the equation by dividing both sides by \(P_1\): \(V_1 = 2V_2\) Now, solve for \(V_2\) by dividing both sides by 2: \(V_2 = \frac{V_1}{2}\)

Plug in the given volume and calculate

Now, we can plug in the given initial volume, \(V_1 = 2.10 \times 10^{4} \mathrm{~mL}\), into the equation: \(V_2 = \frac{2.10 \times 10^{4} \mathrm{~mL}}{2}\) And calculate the final volume, \(V_2\): \(V_2 = 1.05 \times 10^{4} \mathrm{~mL}\) Thus, the new volume of the gas when the pressure is doubled at constant temperature is \(1.05 \times 10^{4} \mathrm{~mL}\).

Key concepts

Understanding Chemistry and Its Principles

Chemistry is the scientific discipline concerned with the properties, composition, and transformation of substances. The fundamental principles of chemistry are essential for explaining the physical and chemical behavior of matter. An understanding of chemistry helps us to explain phenomena from the microscale of atoms and molecules to the macroscale of materials we encounter in everyday life.

In the context of the exercise, we're exploring a chemical principle that involves gases, which are one of the states of matter. Gases are characterized by their ability to fill any container, change shape, and expand infinitely. They consist of particles that move very quickly and are held together by very weak forces. This leads to interesting relationships between their pressure, volume, and temperature, which are described by the gas laws.

Exploring Gas Laws

Gas laws are a set of scientific laws that describe the behavior of gases and their interactions with changes in pressure, temperature, and volume. Boyle's Law, in particular, is a fundamental piece of the puzzle. It's an inverse relationship which means that if you increase the pressure of a gas (and keep the temperature constant), the volume will decrease, and vice versa.

The exercise provided is a classic example of applying Boyle's Law, which states that the pressure of a gas tends to decrease as the volume increases, as long as the temperature remains constant. By understanding this law, students can predict the behavior of gases under different conditions, which is a vital skill in various scientific fields, from chemistry to physics and engineering.

Understanding the Pressure-Volume Relationship

The pressure-volume relationship is a core aspect of Boyle's Law. It tells us how a gas will respond to changes in volume and pressure. As we've seen, doubling the pressure exerted on a gas (while maintaining the same temperature) will halve its volume, illustrated by the equation from Boyle's Law, \(P_1V_1 = P_2V_2\).

To make sense of this in real-world terms, imagine a syringe filled with air; if you push the plunger halfway in, you're doubling the pressure on the air inside. According to Boyle's Law, the volume of that air will reduce to half as long as the temperature doesn't change. This is a perfect analogy to help visualize what's happening at the molecular level in the problem we've explored. By mastering the concept of the pressure-volume relationship, students can anticipate the outcomes of a variety of experiments and practical applications involving gases.