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.

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.

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.

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.

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.