Question

Sketch a graph of pressure versus Celsius temperature, assuming volume is constant. Label the vertical axis \(\mathrm{P}\) and the horizontal axis \(t\left({ }^{\circ} \mathrm{C}\right) .\) Assume the Celsius temperature approaches zero at the origin.

Short answer

Sketch a straight line starting from \( (0, P_0) \) with a positive slope on a graph with axes labeled \( t \left(^{\circ} \mathrm{C}\right) \) and \( P \).

Step-by-step solution

Understand the Relationship

When the volume is constant, the pressure of a gas is directly proportional to its temperature in Kelvin. This is described by Gay-Lussac's law, which can be expressed as: \( P \propto T \). Since the question is about Celsius, we use the conversion \( T(K) = t(°C) + 273.15 \).

Set Up the Equation

Express pressure \( P \) as a linear function of the Celsius temperature \( t \): \[ P = k(t + 273.15) \] where \( k \) is a constant that depends on the specific gas and conditions.Since we are graphing, only the linearity matters, not the exact value of \( k \).

Identify the Graph Features

The graph will be a straight line with a positive slope, since pressure increases with temperature. The intercept on the temperature axis occurs when \( P = 0 \), i.e., when \( t = -273.15 \). However, on our graph starting from zero temperature, this intercept is not plotted.

Sketch the Graph

Draw the horizontal axis labeled as \( t \left(^{\circ} \mathrm{C}\right) \) and the vertical axis as \( P \). Starting from the origin (where \( t = 0 \), a chosen point for convenience), sketch a straight line that rises as it moves to the right.

Label the Graph

Indicate that the line extends in the positive \( t \) direction, continuing upwards as temperature increases. Label critical points, such as the origin \( (0, P_0) \), where \( P_0 \) is the pressure at 0 °C, for illustrative purposes.

Key concepts

Pressure-Temperature Relationship

The relationship between pressure and temperature is a fundamental principle in chemistry, known as Gay-Lussac's Law. This law states that when the volume of a gas is held constant, the pressure of the gas is directly proportional to its temperature, but this is true when the temperature is measured in Kelvin. In mathematical terms, the relationship is expressed as:

• \( P \propto T \) where \( T \) is the temperature in Kelvin.
• With a little rearrangement, we can express this as \( P = kT \), where \( k \) is a constant specific to the situation and the gas involved.

This implies that as the temperature of a gas increases, the pressure also increases, provided that the volume remains unchanged. Conversely, if the temperature decreases, the pressure would decrease as well. This direct proportionality can be easily observed in everyday phenomena, such as when heating a sealed container on a stove, causing the pressure inside to increase noticeably as the temperature rises.

Celsius and Kelvin Conversion

Temperature conversion is crucial when dealing with gas laws, especially when transitioning between Celsius and Kelvin scales. The Kelvin scale is the standard for thermodynamic temperature measurement in scientific contexts. A simple conversion formula allows one to convert Celsius temperatures to Kelvin:

• \( T(K) = t(°C) + 273.15 \)

This equation highlights the absolute nature of the Kelvin scale, where zero Kelvin is absolute zero, the lowest physically possible temperature. For graphing the pressure-temperature relationship using Gay-Lussac’s Law, it’s often necessary to calculate temperatures in Kelvin to establish the starting point of zero pressure, theoretically at \( -273.15° C \), which is absolute zero.

Understanding this conversion ensures accurate visualization and calculation when dealing with gas behaviors under varied thermal conditions. Therefore, keeping track of both Celsius and Kelvin units is vital when applying Gay-Lussac's Law, especially in practical and experimental settings where precision is key.

Graph Sketching in Chemistry

Graph sketching in chemistry is a vital skill, particularly for illustrating relationships such as the one between pressure and temperature. Understanding how to plot these relationships helps visualize experimental data and theoretical principles. When sketching a graph of pressure versus Celsius temperature for a gas at constant volume, there are important steps to follow:

• Label the axes correctly: The horizontal axis should represent temperature \( t(°C) \), and the vertical axis should represent pressure \( P \).
• Determine key features: Since pressure increases linearly with temperature (based on the equation \( P = k(t + 273.15) \)), the graph will be a straight line with a positive slope, beginning at some initial pressure \( P_0 \) at \( 0 °C \).
• Identify intercepts and important points: While the graph usually starts at zero Celsius for convenience, it theoretically would intercept the temperature axis at \( -273.15° C \). However, in a practical sense, this is not plotted for simplicity.
• Sketch the graph: Start the line from the origin and allow it to rise towards the right, indicating increasing pressure with temperature.

Accurate graphing not only helps in understanding gas behaviors but also is critical in predicting outcomes in real-world applications where these relationships are explored, such as in engineering or research.