Question

A fixed amount of oxygen gas is held in a 1.00-L tank at a pressure of 3.50 atm. The tank is connected to an empty 2.00-L tank by a tube with a valve. After this valve has been opened and the oxygen is allowed to flow freely between the two tanks at a constant temperature, what is the final pressure in the system?

Short answer

The final pressure in the system after the gas is redistributed is \(1.17 atm\).

Step-by-step solution

Identify the given information

We are given the initial pressure (P1) and volume (V1) of the first tank and the volume (V2) of the second tank: - P1 = 3.50 atm - V1 = 1.00 L - V2 = 2.00 L We need to find the final pressure (P2) when the oxygen evenly distributes between the two tanks.

Calculate the initial moles of gas

Since the tanks are connected, the moles of gas are conserved, so we can find the initial moles of gas in the first tank by using the Ideal Gas Law equation: PV = nRT Where: - P1: initial pressure of the first tank = 3.50 atm - V1: initial volume of the first tank = 1.00 L - n1: moles of gas in the first tank - R = 0.0821 L atm / (K mol): the ideal gas constant - T: temperature We don’t know the value of the temperature (T), but this equation will help us find the relation between n1, n2 and T later on.

Apply Boyle's Law

Since the temperature remains constant, we can apply Boyle's Law (P1V1 = P2V2) to find the final pressure: 3.50 atm × 1.00 L = P2 × (1.00 L + 2.00 L) Once we distribute the pressure in both tanks, the total volume (V2) will be the sum of the volumes of both tanks.

Solve for the final pressure

Now, we will solve for the final pressure (P2) in the system: 3.50 atm × 1.00 L = P2 × 3.00 L Divide both sides by 3.00 L to find P2: P2 = 3.50 atm × 1.00 L / 3.00 L P2 = 1.17 atm The final pressure in the system after the gas is redistributed is 1.17 atm.

Key concepts

Boyle's Law

Boyle's Law is a fundamental principle in chemistry and physics that describes the relationship between the pressure and volume of a gas when its temperature and amount are kept constant. Named after the physicist Robert Boyle, who first observed the relation in the 17th century, it states that the volume of a given amount of gas held at a constant temperature varies inversely with the applied pressure when the temperature and mass are constant. In other words, if you increase the pressure on a gas, its volume decreases, and vice versa, as long as the quantity of gas and the temperature remain the same.

This can be expressed mathematically as:
\( P_1V_1 = P_2V_2 \).
Where \(P_1\) and \(V_1\) represent the initial pressure and volume, respectively, and \(P_2\) and \(V_2\) represent the final pressure and volume. The product of the initial pressure and volume will always equal the product of the final pressure and volume. This law enables us to perform calculations in many practical scenarios, such as when a gas is transferred from one container to another, as seen in the provided exercise.

Gas Pressure and Volume Relationship

Understanding the intrinsic relationship between gas pressure and volume is crucial for various applications in science and engineering. This relationship is governed by the behavior of gas molecules when confined in a space. When gas is compressed into a smaller volume, the molecules have less space to move around, leading to more frequent collisions with the container’s walls. The more collisions there are, the higher the pressure. Conversely, when a gas is allowed to expand into a larger volume, the molecules collide less frequently, resulting in a lower pressure.

As an ideal model, gases are assumed to have perfectly elastic collisions and the molecules occupy no volume. This simplification is essential for calculations using Boyle's Law as it provides a way to understand how changing one component (pressure or volume) will affect the other, assuming all other variables are constant. In the exercise, Boyle's Law helps predict the final pressure after a fixed amount of gas is allowed to expand from a smaller volume (1 L) into a larger combined volume (3 L).

Moles of Gas in Chemistry

A 'mole' is a unit of measurement used in chemistry to represent a specific number of particles, typically atoms or molecules. One mole is equal to Avogadro’s number, approximately \(6.022 \times 10^{23}\) entities. In gas law calculations, the quantity of a gas is often expressed in moles because it directly relates to the number of molecules present, irrespective of the type of gas.

The ideal gas law, \(PV = nRT\), relates the pressure (P), volume (V), number of moles (n), and temperature (T) of an ideal gas with the universal gas constant (R). From the ideal gas equation, one can determine that if the pressure and volume are known, and the temperature and the gas constant are constants, then the number of moles of the gas can also be calculated. In the exercise given, though the exact moles of oxygen are not calculated, they are constant throughout the process because they are conserved. This means that despite changes in pressure and volume, the amount of gas in moles remains the same when the valve between the two tanks is opened.