Question

State how the pressure of a gas at constant volume and temperature depends on the amount of gas present both with a mathematical relationship and an English statement. Also, make a graph that demonstrates the mathematical relationship.

Short answer

The pressure of a gas at constant volume and temperature is directly proportional to the amount of gas present, following the mathematical relationship \(P = kn\), where k is a constant depending on the gas constant (R), temperature (T), and volume (V). This means that as the amount of gas increases, the pressure increases proportionally, and as the amount of gas decreases, the pressure decreases proportionally. A graph demonstrating this relationship would have pressure (P) on the y-axis and the amount of gas (n) on the x-axis, resulting in a straight, diagonal line upwards from the origin, indicating the direct proportionality between pressure and the amount of gas present.

Step-by-step solution

Recall the Ideal Gas Law

The Ideal Gas Law is given by the formula: \(PV = nRT\), where P is the pressure, V is the volume, n is the amount of gas in moles, R is the gas constant, and T is the temperature. In this exercise, we're interested in the relationship between pressure (P) and the amount of gas present (n), while keeping volume (V) and temperature (T) constant.

Identify the mathematical relationship between pressure and amount of gas present

Since volume (V) and temperature (T) are constant, we can rewrite the Ideal Gas Law as follows: \(P = \frac{nRT}{V}\) Since R, T, and V are constants, we can define a new constant \(k = \frac{RT}{V}\), so the relationship becomes: \(P = kn\) This is the mathematical relationship between pressure and the amount of gas present.

Provide an English statement of the relationship

The pressure of a gas at constant volume and temperature is directly proportional to the amount of gas present. This means that if the amount of gas increases, the pressure increases proportionally, and if the amount of gas decreases, the pressure decreases proportionally.

Create a graph that demonstrates the mathematical relationship

To graph the relationship \(P = kn\), we can plot pressure (P) on the y-axis and the amount of gas (n) on the x-axis. Since this is a linear relationship, the graph will be a straight line with a slope equal to the constant k, which represents the proportionality between pressure and the amount of gas present, and passing through the origin (0, 0). To create the graph, you can follow these steps: 1. Label the x-axis as "Amount of gas (n)" and the y-axis as "Pressure (P)". 2. Choose a value for the constant k (for example, k = 2) and calculate the pressure for different amounts of gas. 3. Plot the points on the graph and connect them with a straight line. 4. The resulting line should be a straight, diagonal line upwards from the origin, indicating a direct proportionality between the pressure of a gas and the amount of gas present at constant volume and temperature.

Key concepts

Pressure of a Gas

Pressure is a measure of the force exerted by the gas particles as they collide with the walls of their container. Imagine a balloon filled with air; the air molecules inside are constantly bumping against the walls of the balloon. Each collision contributes to the force against the walls, which we perceive as pressure.

In the context of the Ideal Gas Law, pressure is denoted by the letter \( P \). It is one of the main variables in the equation \( PV = nRT \), where \( P \) stands for pressure, \( V \) for volume, \( n \) for the number of moles of gas, \( R \) for the gas constant, and \( T \) for temperature. This equation shows how pressure relates to the other aspects of a gas under a given set of conditions. Understanding pressure helps us predict how a gas will behave in different scenarios.

Some key points to remember about the pressure of a gas include:

• Pressure increases when more gas molecules are present in a confined space.
• Pressure decreases if the number of gas molecules is reduced.
• Pressure is affected by changes in temperature and volume, but in this exercise, those are held constant.

Amount of Gas

The amount of gas, given in moles and represented by \( n \), indicates the number of gas molecules present. A mole is a standard scientific unit that facilitates the counting of particles like atoms or molecules, usually extremely numerous in gases.

When we refer to increasing the amount of gas, we're talking about adding more moles of gas into a container. According to the Ideal Gas Law, as \( n \) increases, so does the pressure if the volume and temperature are constant.

Here's why the amount of gas matters:

• Adding more moles increases the frequency of collisions between gas molecules and the container walls, hence raising the pressure.
• If gas escapes from a container, \( n \) decreases, leading to less frequent collisions and a drop in pressure.
• The relationship between \( n \) and pressure is simple and direct in a closed system at consistent temperature and volume.

Direct Proportionality

Direct proportionality is a straightforward mathematical relationship. It means that two variables increase or decrease at the same rate. In our scenario, pressure is directly proportional to the amount of gas. This relationship is described by the equation \( P = kn \).

Since \( k \) (a constant) represents \( \frac{RT}{V} \), it stays the same if the temperature and volume do not change. As such, the pressure and amount of gas maintain a linear relationship.

What does direct proportionality mean for gas behavior?

• If you double the amount of gas (\( n \)), the pressure (\( P \)) will also double, assuming other factors stay constant.
• The graph of this relationship is a straight line passing through the origin, exhibiting a linear increase.
• This proportionality simplifies understanding and predicting the effects of changing gas quantities on pressure within a stable environment.