Question

A different civilization on a distant planet has developed a new temperature scale based on ethyl alcohol. The freezing point of ethyl alcohol \(\left(-117^{\circ} \mathrm{C}\right)\) is designated as \(0^{\circ} \mathrm{J},\) and its boiling point \(\left(78^{\circ} \mathrm{C}\right)\) is designated as \(100^{\circ} \mathrm{J}\). Assuming that the relationship between \({ }^{\circ} \mathrm{C}\) and \({ }^{\circ} \mathrm{J}\) is linear, (a) draw a graph of the line using the data above. (b) what is the slope of the line? (c) what is the \(y\) -intercept of the line? (d) Write an equation to convert \({ }^{\circ} \mathrm{J}\) to \({ }^{\circ} \mathrm{C}\).

Short answer

Answer: The equation to convert from Celsius to the new temperature scale based on ethyl alcohol is: \(J = \frac{100}{195}C + \frac{11700}{195}\).

Step-by-step solution

Understand the given data and plot the points

We are given two points on the Celsius (C) scale that correspond to specific points on the new temperature scale (J). The freezing point of ethyl alcohol is (-117°C, 0°J) and its boiling point is (78°C, 100°J). Plot these points on a graph with C on the x-axis and J on the y-axis.

Determine the slope of the line

The slope of the line (m) is the change in the y-coordinate divided by the change in the x-coordinate between two points. So, considering the points (-117°C, 0°J) and (78°C, 100°J), we have: \(m = \frac{100 - 0}{78 - (-117)} = \frac{100}{195}\)

Determine the y-intercept of the line

Since we know one of the points on the line is (-117°C, 0°J), we can use this point and the slope we found in step 2 to determine the y-intercept (b). Remember that the equation of a line in slope-intercept form is y = mx + b, where y is the J-value and x is the C-value. \(0 = \frac{100}{195}(-117)+b\) Solve for b: \(b = \frac{11700}{195}\)

Write the equation to convert from J to C

Now that we know the slope and the y-intercept, we can write the equation to convert J to C. The equation of the line in slope-intercept form is: \(J = \frac{100}{195}C + \frac{11700}{195}\)

Answer the questions (a)-(d)

(a) Draw a graph of the line using the data above: Plot the points (-117°C, 0°J) and (78°C, 100°J) on a graph with the C-axis and the J-axis, and draw a line connecting them. (b) The slope of the line is \(\frac{100}{195}\). (c) The y-intercept of the line is \(\frac{11700}{195}\). (d) The equation to convert from J to C is: \(J = \frac{100}{195}C + \frac{11700}{195}\).

Key concepts

Ethyl Alcohol Freezing and Boiling Points

Ethyl alcohol, also known as ethanol, plays a significant role in temperature calibration due to its well-defined freezing and boiling points. In the context of the given exercise, the freezing point of ethyl alcohol is marked at \( -117^\circ \text{C} \) and its boiling point at \( 78^\circ \text{C} \). These two points serve as reference points for the formation of a new temperature scale—designated as degrees J—on a distant planet. Understanding these fixed points is crucial for various applications, including the manufacture of thermometers, and it provides a consistent basis for temperature scale conversion.

It's intriguing to note how different substances with unique freezing and boiling points can be used to create various temperature scales. In the case of ethyl alcohol, its properties allow it to remain in liquid form across a wide range of temperatures, making it useful in both laboratory and industrial settings. Moreover, the sharp freezing and boiling points of ethyl alcohol ensure precise temperature conversion and measurement for scientific purposes.

Linear Relationship in Chemistry

In chemistry, a linear relationship refers to a direct proportionality between two variables, where a change in one variable results in a predictable and consistent change in the other. This is seen in the exercise where the temperature scale conversion between the Celsius scale and the new J scale is assumed to be linear. A linear relationship implies that for each degree change in Celsius, there is a corresponding uniform change in degrees J.

Recognizing linear relationships is foundational in chemistry because it allows for the creation of conversion formulas, like the one needed for the new J temperature scale. This approach can also be applied to other chemical properties, such as pressure and volume relationships in gases (Boyle's Law) or concentration and absorbance in spectrophotometry (Beer-Lambert Law). Knowing that a linear relationship exists enables scientists to predict outcomes and establish precise measurements.

Plotting Temperature Conversion Graph

Plotting a temperature conversion graph is an effective method to visualize the relationship between two different temperature scales. The graph typically displays one temperature scale on the x-axis and the other on the y-axis. To plot such a graph, as detailed in the exercise, two points are required — in this case, the freezing and boiling points of ethyl alcohol. Plotting these points (\( -117^\circ \text{C}, 0^\circ \text{J} \) and \( 78^\circ \text{C}, 100^\circ \text{J} \) respectively) allows us to draw a straight line connecting them, indicating the linear relationship between the Celsius and J scales.

For students, constructing a temperature conversion graph not only aids in visual learning but also solidifies their understanding of how different scales correlate. Additionally, the graphical representation can be essential in interpreting temperatures without the need for complex calculations, thereby confirming the utility of visual tools in comprehending scientific concepts.

Slope-Intercept Form

The slope-intercept form is a mathematical equation of a straight line written as \( y = mx + b \), where \( m \) is the slope of the line, and \( b \) is the y-intercept. This format is incredibly useful for representing linear relationships, including those in chemistry. In the given exercise, the equation to convert from the Celsius scale (\( C \) values) to the new J scale (\( J \) values) can be expressed in the slope-intercept form.

The slope \( m \) represents the rate at which the y-values (degrees J) change for every unit change in x-values (degrees C). Meanwhile, the y-intercept \( b \) represents the value of J when the Celsius temperature is zero. Understanding how to use and manipulate the slope-intercept form enables students to create their own conversion equations, predict values, and understand the rate of change within the context of temperature or any other linear relationship.