Question

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

Short answer

The graph is a straight line through the origin with a positive slope, showing that volume increases linearly with temperature.

Step-by-step solution

Understand the Relationship

According to Charles's Law, at constant pressure, the volume of a gas is directly proportional to its temperature in Kelvin. Mathematically, this is expressed as \( V \propto T \) or \( V = kT \), where \( k \) is a constant. This implies that as temperature increases, volume increases linearly.

Determine the Equation Format

Given the relationship \( V = kT \), the equation represents a straight line passing through the origin (0,0), assuming a Kelvin temperature scale where negative values aren't possible. Therefore, the graph should be a straight line rising from the origin.

Sketch the Graph

Draw a graph with the vertical axis labeled as \( V \) (volume) and the horizontal axis labeled as \( T(K) \) (temperature in Kelvin). Begin the line at the origin (since volume is zero at zero Kelvin) and sketch a straight line extending upwards and to the right indicating a positive slope. This line shows the increase in volume with temperature.

Add Labels and Key Points

Add labels to crucial points on the axes, noting any known values if provided or relevant (e.g., normal room temperature in Kelvin). Label the curve itself if needed, and ensure the direction of increasing volume is clear.

Key concepts

Volume-Temperature Relationship

The volume-temperature relationship is a fundamental concept in understanding how gases behave under various conditions, specifically when pressure remains constant. Charles's Law captures this idea, stating that the volume of a gas is directly proportional to its temperature, as long as the pressure does not change. This is mathematically formulated as \( V \propto T \), which simplifies to \( V = kT \), where \( V \) is the volume, \( T \) is the temperature in Kelvin, and \( k \) is a constant.

### How It Works

• As the temperature increases, the molecules within the gas gain more energy, causing them to move faster and occupy more space, thus increasing the volume.
• Conversely, when the temperature decreases, the molecules lose energy, slow down, and the volume decreases accordingly.
• This relationship is linear, meaning if you were to graph the change, it would form a straight line with a positive slope, beginning from the origin.

This understanding is crucial not just for academic purposes but also for practical applications, such as predicting how a gas will behave with temperature changes in real-world scenarios.

Gas Laws

Gas laws describe how gases react to changes in temperature, volume, and pressure. Charles's Law is one among several that help predict and explain the behavior of gaseous materials. Each gas has its specific set of rules identified under these laws, usually covering changes involving volume, temperature, and pressure.

### Key Points of Gas Laws

• **Boyle's Law:** States that at a constant temperature, the pressure of a gas is inversely proportional to its volume.
• **Charles's Law:** Relates the volume and temperature of a gas, highlighting direct proportionality for constant pressure.
• **Avogadro's Law:** Indicates that equal volumes of gases at the same temperature and pressure contain an equal number of molecules.

Together, they form the combined gas law or the ideal gas law, broadly used in scientific investigations. Each law highlights a different relationship, yet they work together to give a comprehensive picture of gaseous behavior under different conditions.

Kelvin Temperature Scale

The Kelvin temperature scale is essential in gas law calculations for its treatment of 0 Kelvin as absolute zero—the point where molecular energy is minimal. Unlike Celsius or Fahrenheit scales, Kelvin begins at this absolute zero, eliminating negative temperature values, which correspond to impossible physical states of a gas.

### Importance of Kelvin

• **Zero Point:** It starts from absolute zero, making it ideal for scientific calculations involving gases.
• **Linear Relation:** Ensures that temperature changes directly affect the properties of a gas without the distortions caused by negative values in other scales.
• **Universal Applications:** Many scientific formulas are based on the Kelvin scale, making it a universal standard.

Kelvin is crucial for understanding gas behavior because it aligns perfectly with theoretical models of temperature and energy. It provides a straightforward, unambiguous basis for studying gases, instrumental for systems where precision is key.