Problem 24 - Algebra and Trigonometry with Analytic Geometry

Question

The ideal gas law states that the volume \(V\) that a gas occupies is directly proportional to the product of the number \(n\) of moles of gas and the temperature \(T\) (in \(\mathrm{K}\) ) and is inversely proportional to the pressure \(P\) (in atmospheres). (a) Express \(V\) in terms of \(n, T, P\), and a constant of proportionality \(k\). (b) What is the effect on the volume if the number of moles is doubled and both the temperature and the pressure are reduced by a factor of one-half?

Step-by-Step Solution

Answer

Volume doubles if moles double and temperature, pressure are halved.

Step 1: Understand the Ideal Gas Law

The ideal gas law relates four main variables involved with a gas: the number of moles \(n\), the temperature \(T\), the pressure \(P\), and the volume \(V\). It states that \(V\) is directly proportional to \(n \times T\) and inversely proportional to \(P\). This relationship is mathematically expressed using a constant of proportionality \(k\).

Step 2: Express Volume \(V\) with Proportionality

Given the relationship outlined, we can express the ideal gas law as \(V = k \frac{nT}{P}\). Here, \(k\) is the constant of proportionality which depends on the particular units and assumptions of the system.

Step 3: Analyze Changes to Variables

The problem states that \(n\) is doubled (\(n\to 2n\)), and both \(T\) and \(P\) are reduced by half (\(T \to \frac{T}{2}\) and \(P \to \frac{P}{2}\)). Substitute these changes into the equation derived:

Step 4: Substitute and Simplify

Substitute the changes into the equation: \[V' = k \frac{(2n)\left(\frac{T}{2}\right)}{\left(\frac{P}{2}\right)}\]Simplify the expression:\[V' = k \frac{2n \times T}{P}\]And simplifying further:\[V' = 2(k \frac{nT}{P})\]Hence, the new volume \(V'\) is twice the original volume \(V\).

Step 5: Conclude

The volume is affected by the changes in the initial conditions as described in Step 4. When the number of moles is doubled, and both temperature and pressure are reduced by half, the volume increases by a factor of 2.

Key Concepts

Volume in the Ideal Gas Law

When discussing gases, volume refers to the amount of space that the gas occupies. In the context of the ideal gas law, volume is represented by the symbol \( V \). The ideal gas law helps us understand the behavior of gases under different conditions. It reveals that the volume is influenced by three key factors: the amount of gas (moles), the temperature, and the pressure. The equation representing this relationship is \( V = k \frac{nT}{P} \), where \( k \) is a constant that ensures the relationship holds true based on units and system specifics. This equation shows that as the product of moles and temperature increases, the volume increases, assuming pressure stays constant. Likewise, if pressure increases while other factors remain constant, the volume will decrease. This relationship provides a foundation for predicting how gases will behave in various conditions.

Understanding Proportionality in Gases

Proportionality is a core concept in understanding how the properties of a gas interact. When we say two quantities are "proportional," it means that if one quantity changes, the other changes in a specific way. With the ideal gas law, we have two types of proportionality at play:

• Direct Proportionality: Volume is directly proportional to the product of moles \( n \) and temperature \( T \). This means that if you increase the number of moles or the temperature, the volume will also increase.
• Inverse Proportionality: Volume is inversely proportional to pressure \( P \), meaning that if pressure increases, the volume decreases if all other factors are constant.

This dual proportionality helps us predict how gas volume will change in response to alterations in conditions.

How Temperature Affects Gas Volume

Temperature is a measure of the average kinetic energy of particles in a substance. In the context of gases, as the temperature increases, so does the kinetic energy of the gas particles. This increase in kinetic energy causes the particles to move more vigorously, often leading to an increase in volume.In the ideal gas law \( V = k \frac{nT}{P} \), you've likely noticed that volume \( V \) is directly proportional to temperature \( T \). This implies:

• As temperature increases, for a constant amount of gas and pressure, the volume will increase.
• If temperature decreases, the volume decreases assuming the amount of gas and pressure remain constant.

Understanding this relationship is crucial for studying thermodynamic systems and how gases behave under changing thermal environments.

Pressure and Its Impact on Gas Volume

Pressure is the force exerted by the gas particles colliding with the walls of its container. In terms of the ideal gas law, pressure plays a significant role in defining the volume a gas occupies. According to the equation \( V = k \frac{nT}{P} \), volume is inversely proportional to pressure. From this, we derive:

• Increasing the pressure, while keeping moles and temperature constant, leads to a decrease in volume.
• Conversely, decreasing the pressure allows the volume to expand if moles and temperature are unchanged.

This inverse relationship highlights why gases occupy more space at lower pressures and is fundamental to various applications like inflating balloons or pressure cooking.

Role of Moles in Gas Volume

Moles refer to the amount of substance, specifically the number of atoms or molecules in a gas. In the ideal gas law expression \( V = k \frac{nT}{P} \), \( n \) represents the moles. The more moles of gas present, the larger the volume it will occupy, as long as temperature and pressure are constant. Here’s how moles influence the volume:

• Increasing the moles (amount of gas) increases the volume, assuming temperature and pressure are unchanged.
• Decreasing the moles will lead to a decrease in volume, given constant temperature and pressure.

Understanding the role of moles allows us to conceptualize how adding or removing gas changes the space it occupies, an essential concept in both chemistry and physics.