Question

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

Short answer

The graph is a straight line through the origin, sloping upward to the right.

Step-by-step solution

Understanding the Relationship Between Volume and Temperature

According to Charles's Law, the volume of a gas is directly proportional to its temperature when pressure is constant. This can be represented by the equation: \[ V = k(t + 273.15) \] where \( V \) is the volume, \( k \) is a constant of proportionality, and \( t \) is the temperature in Celsius.

Setting the Origin of the Graph

In the given problem, the Celsius temperature approaches zero at the origin. Thus, the origin (0,0) is where the temperature is \(0^{\circ}C\). Even though this is a simplification, we will assume this origin for the purpose of sketching our graph.

Drawing the Axes and Labeling

Draw two perpendicular lines to form the axes. Label the vertical axis as \( V \) and the horizontal axis as \( t(^{\circ}C) \). The origin will be the point where \( t = 0 \), representing \(0^{\circ}C\).

Sketching the Linear Graph

Because the volume increases linearly with temperature, draw a straight line that passes through the origin. This line should slope upwards to the right, reflecting the direct proportionality between volume and temperature. Any specific point on this graph can be found using a known volume at a specific temperature, following Charles's Law.

Key concepts

Gas Laws

Gas laws describe the behavior of gases and their interactions with pressure, volume, and temperature. A key player in understanding these laws is Charles's Law. Charles's Law states that the volume of a gas is directly proportional to its temperature when pressure remains constant. This means, as the temperature of a gas increases, so does its volume, provided the pressure doesn't change. It is important to remember that temperature must be measured in Kelvin for gas law calculations, as Kelvin is the absolute temperature scale.

• **Charles's Law Formula**: \[ V = k(T) \] where \( V \) is volume, \( k \) is a constant, and \( T \) is the temperature in Kelvin.
• **Proportional Relationship**: More temperature → More volume.
• **Constant Pressure**: This law only applies when no alterations are made to the gas pressure.

Understanding gas laws like Charles's Law helps predict how gases will react under different conditions, such as heating a balloon or using an air pump. These laws form the foundation of thermodynamics and help us explain everyday phenomena.

Volume-Temperature Relationship

The volume-temperature relationship is straightforward under Charles's Law. It addresses how the volume of a gas changes with a change in temperature, provided pressure is constant.
When examining this relationship, you can see that:

• As temperature increases, gas particles move faster, causing them to spread apart and take up more space, hence increasing volume.
• Conversely, when temperature decreases, particles slow down, and the gas contracts, reducing the volume.
• This relationship is linear, meaning if you double the temperature, you double the volume, assuming pressure and the amount of gas remain unchanged.

To calculate how much the volume changes, convert Celsius to Kelvin by adding 273.15, because calculations in gas laws require absolute temperatures. If you start with a temperature below absolute zero ( -273.15°C), the gas would have theoretically zero volume, which practically communicates the need for using Kelvin in calculations.
Knowing this relationship is crucial in contexts where gases are stored or utilized, such as in weather balloons or refrigeration cycles. It helps assure proper and safe functioning by allowing us to anticipate the behavior of gases under thermal variations.

Graphical Representation of Gas Behavior

To visually interpret how a gas behaves under different temperature conditions, a volume versus temperature graph can be very helpful. In this representation, the vertical axis (\( V \)) signifies volume, while the horizontal axis (\( t({}^{\circ}\mathrm{C}) \)) denotes temperature in Celsius.

• A graph following Charles's Law typically shows a straight, upward-sloping line starting from a specific point near the origin, influenced by the assumption regarding temperature conversion.
• When the temperature of the gas nears absolute zero (converted to Kelvin), the volume approaches a minimum, yet in scientific practice, reaching absolute zero is impossible.
• This graph can vary slightly based on calibration, but generally, the direct proportionality will always reflect in the steady incline as temperature and volume increase simultaneously.

Illustrating these relationships graphically aids learners and professionals alike in predicting gas behavior easily. Such graphs are also fundamental in scientific explorations and applications, like calibrating instruments and designing containers for gas storage, by providing an intuitive grasp of fundamental theoretical concepts.