Problem 9 - Basic Engineering Mathematics

Question

In an experiment on Charles's law, the value of the volume of gas, \(V \mathrm{~m}^{3}\), was measured for various temperatures \(T^{\circ} \mathrm{C}\). Results are shown below. $$ \begin{array}{|l|llllll|} \hline V \mathrm{~m}^{3} & 25.0 & 25.8 & 26.6 & 27.4 & 28.2 & 29.0 \\ T^{\circ} \mathrm{C} & 60 & 65 & 70 & 75 & 80 & 85 \\ \hline \end{array} $$ Plot a graph of volume (vertical) against temperature (horizontal) and from it find (a) the temperature when the volume is \(28.6 \mathrm{~m}^{3}\), and (b) the volume when the temperature is \(67^{\circ} \mathrm{C}\).

Step-by-Step Solution

Answer

(a) The temperature is approximately 83°C; (b) the volume is approximately 26 m³.

Step 1: Understand the Relationship

Charles's Law states that the volume of gas is directly proportional to its temperature, assuming pressure is constant. This means that if we plot volume against temperature, the points should form a straight line.

Step 2: Organize the Data

Reorganize the data given: Temperatures ( T^{ ext{°C}} ) are 60, 65, 70, 75, 80, and 85. Volumes ( V m³) corresponding to these temperatures are 25.0, 25.8, 26.6, 27.4, 28.2, and 29.0.

Step 3: Plot the Graph

On graph paper or using graph software, place the temperature ( T ) on the horizontal (x-axis) and the volume ( V ) on the vertical (y-axis). Plot the data points: (60, 25.0), (65, 25.8), (70, 26.6), (75, 27.4), (80, 28.2), and (85, 29.0).

Step 4: Draw the Line of Best Fit

Once all points are plotted, draw a straight line that best represents the data points. It may not pass through every point, but it should reflect the overall trend of the data.

Step 5: Determine When Volume is 28.6 m³

Locate 28.6 on the vertical axis (volume), and draw a horizontal line from this point until it intersects the line of best fit. From the point of intersection, draw a vertical line downwards to the horizontal axis (temperature) and record the temperature value. This is the temperature when the volume is 28.6 m³.

Step 6: Determine Volume at 67°C

Find 67°C on the horizontal axis and draw a vertical line upwards until it intersects the line of best fit. From the intersection point, draw a horizontal line towards the vertical axis (volume) to record the volume at 67°C.

Key Concepts

Gas Laws

Gas laws are fundamental principles that explain how the properties of gases, such as temperature, volume, and pressure, are interrelated. One of the key gas laws is Charles's Law. This law is part of a broader group of gas laws that also includes Boyle's Law, Avogadro's Law, and Gay-Lussac's Law.

• Charles's Law focuses on the relationship between volume and temperature, setting aside pressure as a constant. According to this law, the volume of a gas is directly proportional to its absolute temperature, provided that the pressure remains constant.
• An increase in temperature will lead to an increase in volume, and conversely, a decrease in temperature will lead to a decrease in volume.

Understanding gas laws is crucial because they help predict how gases will behave under various environmental changes. This knowledge is applied across numerous fields, from chemistry experiments to real-world applications like weather forecasting and engineering.

Volume-Temperature Relationship

The volume-temperature relationship is the core of Charles's Law. It tells us how the volume of a gas changes as its temperature changes. In mathematical terms, Charles's Law is represented as:
\[ V \propto T \] Which can be rewritten as:
\[ \frac{V}{T} = k \]
where \( V \) is the volume, \( T \) is the temperature in Kelvin, and \( k \) is a constant for a given amount of gas at constant pressure.

• To get the temperature in Kelvin, add 273.15 to the Celsius temperature.
• This equation shows that if one variable increases, the other must increase as well to maintain the proportionality constant \( k \).

In the exercise, we see this relationship by plotting volume against temperature. A straight line indicates that the proportional relationship holds true under the conditions of the experiment with pressure held constant.

Graph Plotting

Graph plotting helps visually understand relationships between different variables. In studying Charles's Law, we use graphs to see how volume and temperature of a gas interact. Let's break down the process:

• Axes Setup: Temperature is typically on the horizontal x-axis, and volume is on the vertical y-axis.
• Plotting Points: Begin by marking the points given in the exercise, for example, (60, 25.0) and (85, 29.0). These points are then connected, ideally creating a straight line.
• Line of Best Fit: If the points do not perfectly align, draw a "line of best fit" that best represents the trend of the data. This line allows you to make predictions or estimates of values not explicitly given in your data set.

By plotting these kinds of graphs, you can visually identify direct relationships and predict unknown values, which is the aim in tasks such as this exercise.

Temperature Measurement

Proper temperature measurement is critical for correctly applying Charles's Law. Temperature affects the volume of a gas, so it must be accurately measured in experiments. A few key points include:

• Measurement Tools: Thermometers or thermostats are typically used to measure temperatures. Digital thermometers provide higher accuracy.
• Scale: In most scientific calculations, temperature should be converted from Celsius to Kelvin by adding 273.15. This conversion is essential because the Kelvin scale starts at absolute zero, the theoretical point where particles stop moving.
• Consistency: Ensure that the temperature measurements are consistent to maintain the accuracy and reliability of the experimental outcomes.

Adhering to these principles ensures that calculations based on gas laws are as precise as possible, leading to better understanding and more accurate predictions.