Question

Sketch a graph of pressure versus Kelvin temperature, assuming volume is constant. Label the vertical axis \(\mathrm{P}\) and the horizontal axis \(\mathrm{T}(\mathrm{K})\).

Short answer

The graph is a straight line through the origin, showing pressure directly proportional to Kelvin temperature.

Step-by-step solution

Understand the Relationship

When the volume is constant, pressure and temperature are related by Gay-Lussac's Law which states that pressure is directly proportional to temperature. Mathematically, this is expressed as \( P \propto T(K) \).

Set Up the Graph Axes

Draw the horizontal axis and label it \( T(K) \) for temperature in Kelvin. Draw the vertical axis and label it \( P \) for pressure.

Determine the Graph Shape

Since pressure is directly proportional to temperature, the graph should be a straight line that passes through the origin (0,0) because a temperature of 0 K implies a pressure of 0 as well.

Plot Points on the Graph

Choose a few points based on arbitrary temperature values. For example, at \( T = 100 \) K, if \( P = 2 \) units, then at \( T = 200 \) K, \( P \) would be 4 units, and so on, intercalating through other points that maintain the direct proportion.

Draw the Line

Connect the plotted points with a straight line starting from the origin. This line represents the direct relationship between pressure and temperature when volume is constant.

Label and Finalize

Ensure the line is labeled appropriately with 'Pressure vs. Kelvin Temperature'. Mark and label the axes with values and units, confirming that the graph reflects the direct proportionality.

Key concepts

Gay-Lussac's Law

Gay-Lussac's Law is a fundamental concept in chemistry that describes the relationship between the pressure and temperature of a gas, while keeping the volume constant. This law is particularly useful in understanding how gases behave under different conditions, which is critical in fields such as thermodynamics and physical chemistry.
According to Gay-Lussac's Law, the pressure exerted by a gas is directly proportional to its temperature in Kelvin. This can be mathematically expressed as \( P \propto T(K) \), where \(P\) is the pressure and \(T(K)\) is the temperature in Kelvin. This means that if the temperature of a gas increases, its pressure also increases, provided the volume does not change.
This relationship is extremely useful when analyzing systems under controlled conditions. For example, suppose pressure is measured in atmospheres and temperature in Kelvin; if the temperature of a contained gas is doubled, the pressure will also double. This characteristics of linearity and direct proportionality make Gay-Lussac's Law a powerful tool for predicting the behavior of gases under thermal fluctuations.

Pressure-Temperature relationship

Understanding the pressure-temperature relationship in gases is essential for predicting how changes in temperature affect gas pressure, a core aspect of Gay-Lussac's Law. This relationship is easily demonstrated by the equation:\[\frac{P_1}{T_1} = \frac{P_2}{T_2}\]Where

• \( P_1 \) and \( P_2 \) are the initial and final pressures, respectively
• \( T_1 \) and \( T_2 \) are the initial and final temperatures in Kelvin

This formula implies that any change in the temperature of a gas, while keeping the volume constant, will result in a corresponding and proportional change in its pressure.
Let's consider a practical example. If a gas inside a rigid container is initially at a temperature of 300 K and a pressure of 1 atm, heating the gas to 600 K will increase the pressure to 2 atm, assuming the volume remains unchanged.
This relationship is pivotal in many real-world applications such as pressurized gas storage, pneumatic systems, and even weather balloon operations where accurate predictions of pressure changes are crucial.

Graphing techniques in Chemistry

Graphing is a powerful visual technique used in chemistry to illustrate the relationships between different variables. In this particular exercise, we are visualizing the relationship between pressure and temperature for a gas at constant volume.
To create a graph for Gay-Lussac's Law:

• Draw two perpendicular axes on graph paper or digital graphing tools.
• Label the horizontal axis as \( T(K) \), representing temperature in Kelvin.
• Label the vertical axis as \( P \), representing pressure.

Once the axes are set, the next step involves plotting points that correspond to the different temperature and pressure values, maintaining their direct proportionality. Start by plotting the origin point (0,0) which illustrates that at absolute zero temperature (0 K), the pressure is zero.
Continuing with selected temperature values, draw a straight line through the plotted points to show that as temperature increases, pressure increases linearly. This straightforward linear relationship offers a clear visual representation of Gay-Lussac's Law.
Understanding how to graph these relationships not only aids in comprehending the theoretical aspects but also enhances the ability to predict and interpret real-world gas behavior under varying temperature conditions.