Question

Draw a qualitative graph to show how the first property varies with the second in each of the following (assume 1 mole of an ideal gas and \(T\) in kelvin). a. \(P V\) versus \(V\) with constant \(T\) b. \(P\) versus \(T\) with constant \(V\) c. \(T\) versus \(V\) with constant \(P\) d. \(P\) versus \(V\) with constant \(T\) e. \(P\) versus \(1 / V\) with constant \(T\) f. \(P V / T\) versus \(P\)

Short answer

a. The graph of PV versus V with constant T will be a straight line with a positive slope. b. The graph of P versus T with constant V will be a straight line with a positive slope. c. The graph of T versus V with constant P will be a straight line with a positive slope. d. The graph of P versus V with constant T will be a downward curve (hyperbola). e. The graph of P versus 1/V with constant T will be a straight line with a positive slope. f. The graph of PV/T versus P will be a horizontal straight line with no slope.

Step-by-step solution

a. Graph of PV versus V with constant T

In this scenario, temperature is constant, and we need to plot the relation between PV and V. From the ideal gas law, we know that PV = nRT. Since temperature is constant, we relate PV with V as follows: PV = Constant We can conclude that the graph will be a straight line with a positive slope.

b. Graph of P versus T with constant V

In this scenario, we need to plot the relationship between pressure and temperature while keeping the volume constant. By rearranging the ideal gas law equation, we get: P = (nR/V)T When volume is constant, the relation between P and T becomes directly proportional. Thus, the graph will be a straight line with a positive slope.

c. Graph of T versus V with constant P

In this scenario, we will plot the relationship between temperature and volume while keeping the pressure constant. Rearranging the ideal gas law equation, we get: T = (PV)/(nR) When pressure is constant, the relation between V and T becomes directly proportional. The graph will be a straight line with a positive slope.

d. Graph of P versus V with constant T

In this scenario, we need to plot the relationship between pressure and volume while keeping the temperature constant. By rearranging the ideal gas law equation, we get: P = (nRT)/V When temperature is constant, the relation between P and V becomes inversely proportional. Thus, the graph will be a downward curve known as a hyperbola.

e. Graph of P versus 1/V with constant T

In this scenario, we need to plot the relationship between pressure and reciprocal of the volume, keeping the temperature constant. From the ideal gas law, we know that P = (nRT)/V. Rewriting this equation in terms of 1/V, we get: P = (nR)T(V^-1) In this case, pressure and the reciprocal of the volume are directly related. Therefore, the graph will be a straight line with a positive slope.

f. Graph of PV/T versus P

In this scenario, we are asked to plot the relationship between (PV/T) and pressure. From the ideal gas law, we know that PV = nRT. If we divide both sides by T, we get: PV/T = nR Since we are considering one mole of an ideal gas, n = 1. Therefore: PV/T = R Since R is a constant, the graph of PV/T versus P will be a horizontal straight line with no slope.

Key concepts

Pressure-Volume Relationship

The relationship between pressure and volume is elegantly described by Boyle's Law. When the temperature is held constant, the product of pressure (P) and volume (V) for a gas remains constant. This means that as the volume of the gas decreases, the pressure increases, and vice versa. The relationship can be represented by the equation\[ PV = ext{constant} \]This is an example of an inverse relationship. If you were to draw a graph of pressure versus volume while keeping the temperature constant, you would see a downward curve known as a hyperbola. This representation helps in understanding that pressure and volume change in opposite directions under constant temperature conditions. Additionally, if we plot pressure against the reciprocal of volume (1/V), the graph becomes a straight line.To sum up:

• Increased volume results in decreased pressure (and vice versa).
• The graph of pressure versus volume is a hyperbola.
• The graph of pressure versus 1/V is a straight line.

Temperature-Volume Relationship

Charles's Law explores the relationship between temperature and volume, providing insight into how gases expand when heated. According to this law, the volume of a gas is directly proportional to its temperature when pressure is held constant. Hence, when you increase the temperature, the gas molecules have more energy and thus, occupy more space. This can be mathematically expressed as:\[ V \propto T \quad \text{or} \quad \frac{V}{T} = ext{constant} \]When graphing volume against temperature at a constant pressure, the graph will be a straight line with a positive slope. In such a graph, as the temperature increases, the volume does too, illustrating their direct proportionality.Key points:

• Higher temperature increases volume.
• The graph is a straight line with a positive slope.
• Direct proportionality between temperature and volume.

Pressure-Temperature Relationship

The relationship between pressure and temperature is explained by Gay-Lussac's Law. When the volume is held constant, the pressure of a gas is directly proportional to its absolute temperature. So, if you increase the temperature of a gas, its pressure also increases provided the volume does not change. This relationship is given by:\[ P \propto T \quad \text{or} \quad \frac{P}{T} = ext{constant} \]When plotting pressure against temperature with a constant volume, you will observe a linear graph with a positive slope. This shows that as the temperature of the gas increases, so does the pressure, showcasing their direct relationship.Summary:

• Pressure increases with an increase in temperature.
• The graph is a straight line with a positive slope.
• A direct relationship between pressure and temperature.

Qualitative Graphs

Qualitative graphs are an excellent tool for visualizing relationships in gas laws. They help establish clear visual interpretations of theoretical concepts like the interdependence of gas properties. - **Pressure vs. Volume graphs**: typically showcase a hyperbola when temperature is constant, highlighting an inverse relationship. - **Temperature vs. Volume graphs**: display a straight line when pressure is constant, illustrating a direct proportionality. - **Pressure vs. Temperature graphs**: also show a straight line when volume is constant, again displaying direct proportionality. By utilizing these graphs, students and scientists alike can predict how alterations in one property affect others. These visual tools simplify complex ideas into more digestible forms. In essence:

• Help in visualizing complex relationships.
• Show whether properties are directly or inversely related.
• Assist in conceptual understanding of the gas behaviors.

Gas Behavior

Understanding gas behavior is crucial in comprehending various natural and industrial processes. Gas laws encapsulate the fundamental aspects influencing how gases behave under different conditions.

• Boyle’s Law explains that at constant temperature, the pressure and volume of a gas are inversely related.
• Charles’s Law states that at constant pressure, the volume and temperature of a gas are directly proportional.
• Gay-Lussac's Law shows that at constant volume, pressure and temperature are directly proportional.

These laws collectively form the Ideal Gas Law expressed as:\[ PV = nRT \]Here, \(n\) is the number of moles, \(R\) is the gas constant, and \(T\) is the temperature in Kelvin. This equation helps predict gas behavior under various scenarios, offering a baseline understanding essential in fields like chemistry and engineering.To recap:

• Describes fundamental gas behaviors.
• Helps predict how gases will react to changes in conditions.
• Essential for scientific and engineering applications.