Question

At conditions of constant temperature and pressure, the volume of a sample of ideal gas is _______ proportional to the number of moles of gas present.

Short answer

The volume of a sample of ideal gas is directly proportional to the number of moles of gas present at constant temperature and pressure, as represented by the formula \(V \propto n\).

Step-by-step solution

Recall the Ideal Gas Law formula

The Ideal Gas Law is given by the equation: \(PV=nRT\), where \(P\) is the pressure of the gas, \(V\) is the volume of the gas, \(n\) is the number of moles of gas present, \(R\) is the ideal gas constant, and \(T\) is the temperature of the gas.

Define constant temperature and pressure

In this problem, we are asked to find the relationship between volume and the number of moles at constant temperature and pressure. This means that both the temperature \(T\) and pressure \(P\) will not change.

Rearrange the Ideal Gas Law equation for constant temperature and pressure

Since \(P\), \(T\), and \(R\) are constant, we can rearrange the Ideal Gas Law equation to express the relationship between the volume \(V\) and the number of moles \(n\). We have: \(PV = nRT\) Since \(P\), \(T\), and \(R\) are constant, we can express their product as a constant, \(k\): \(k = \frac{PV}{n}\) We can now rearrange the equation to find the relationship between \(V\) and \(n\): \(V = \frac{kn}{P}\)

Identify the proportionality between volume and moles

From the rearranged equation, it is clear that the volume of a sample of ideal gas is directly proportional to the number of moles of gas present at constant temperature and pressure: \(V \propto n\) Therefore, the volume of a sample of ideal gas is directly proportional to the number of moles of gas present at constant temperature and pressure.

Key concepts

Directly Proportional Relationship

In the realm of physics and chemistry, understanding how variables interact with one another is crucial. A directly proportional relationship exists when two variables increase or decrease in tandem. In other words, as one variable goes up, the other one does as well; conversely, if one goes down, the other follows. This relationship is graphically represented by a straight line that passes through the origin on a coordinate system.

Using the Ideal Gas Law, we can witness this proportionality at play. To recap, Ideal Gas Law is expressed by the formula \(PV=nRT\). If we hold temperature (\(T\)) and pressure (\(P\)) constant, and we solve for volume (\(V\)), the equation simplifies to \(V = \frac{k}{P} \cdot n\), where \(k\) is a constant that encompasses the product of pressure, temperature, and the ideal gas constant. Notice how the volume \(V\) varies directly with the number of moles of gas \(n\), illustrating a classic directly proportional relationship.

Moles of Gas

The moles of gas concept is foundational in chemistry and critical when discussing the properties of gases. One mole is Avogadro's number of particles, which is approximately 6.022 x 10²³ particles. In the context of the Ideal Gas Law, moles are represented by the symbol \(n\) and provide a quantitative measure of the amount of gas we are dealing with.

For gases, the volume is greatly influenced by how many moles there are. This is particularly important because the Ideal Gas Law assumes that gases consist of a large number of tiny particles that are far apart, and therefore, the volume of the gas is primarily the space that these particles are moving in, not the space they occupy. Increasing the number of moles of gas in a constant volume will increase the pressure, while at constant pressure, increasing the moles will increase the volume, again showcasing the directly proportional relationship between \(V\) and \(n\) at constant temperature and pressure.

Constant Temperature and Pressure

In gas laws, maintaining a constant temperature and pressure is often a condition for deriving straightforward relationships between other variables. Temperature, in thermal physics, is a measure of the average kinetic energy of the particles in a substance; in our case, the gas particles. Pressure, on the other hand, is essentially the force that these particles exert when they collide with the walls of their container. The term 'constant' simply means that these two variables do not change; they're held steady throughout the process.

When we examine the Ideal Gas Law under the lens of constant temperature (\(T\)) and pressure (\(P\)), we end up with simplified scenarios and equations. These fixed conditions allow us to make certain assumptions and therefore, simplify our understanding of how gases behave. The Ideal Gas Law specifically gives us a mathematical model to predict how volume will react to changing numbers of moles when \(T\) and \(P\) are immutable. The stability provided by these constants is essential for the gas to demonstrate a directly proportional relationship between the volume and the moles present.