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Suppose that a new temperature scale has been devised on which the melting point of ethanol \(\left(-117.3^{\circ} \mathrm{C}\right)\) and the boiling point of ethanol \(\left(78.3^{\circ} \mathrm{C}\right)\) are taken as \(0^{\circ} \mathrm{S}\) and \(100^{\circ} \mathrm{S},\) respectively, where \(\mathrm{S}\) is the symbol for the new temperature scale. Derive an equation relating a reading on this scale to a reading on the Celsius scale. What would this thermometer read at \(25^{\circ} \mathrm{C} ?\)

Short Answer

Expert verified
At 25°C, the thermometer reads approximately 72.79°S.

Step by step solution

01

Understand the Problem

We need to create a linear equation to convert temperatures between degrees Celsius (°C) and the new temperature scale (°S), given the melting and boiling points of ethanol. The melting point (-117.3°C) is 0°S, and the boiling point (78.3°C) is 100°S.
02

Determine the Equation Form

The relationship between the two scales is linear, so we'll use the equation of a straight line in the form: \[ S = kC + b \] where \(S\) is the temperature in the new scale, \(C\) is the temperature in Celsius, \(k\) is the slope, and \(b\) is the y-intercept.
03

Calculate Slope \(k\)

Use two known points to calculate the slope: \((C_1, S_1) = (-117.3, 0)\) and \((C_2, S_2) = (78.3, 100)\).Plug these into the slope formula:\[ k = \frac{S_2 - S_1}{C_2 - C_1} = \frac{100 - 0}{78.3 + 117.3} = \frac{100}{195.6} \approx 0.5115 \]
04

Determine the Y-intercept \(b\)

Substitute \(k\) and one of the known points into the equation to solve for \(b\). Using \((C_1, S_1)\):\[ 0 = 0.5115(-117.3) + b \]\[ b = 60 \]
05

Write the Conversion Equation

Substitute \(k\) and \(b\) into the equation formula:\[ S = 0.5115C + 60 \]This is the conversion equation from Celsius to the new temperature scale.
06

Convert 25°C to the New Scale

Plug \(C = 25\) into the conversion equation:\[ S = 0.5115(25) + 60 \] \[ S = 12.7875 + 60 \] \[ S = 72.7875 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Temperature Scales
Temperature scales are systems used to measure and communicate temperature readings. Various scales exist due to different points of reference. Commonly known scales include Celsius (°C), Fahrenheit (°F), and Kelvin (K). Each has specific reference points and different intervals between degrees.

Celsius is based on the freezing point (0°C) and boiling point (100°C) of water at sea level. Fahrenheit sets water's freezing and boiling points at 32°F and 212°F, respectively, while Kelvin is an absolute scale starting at absolute zero (0 K) where no thermal energy remains. These scales allow for scientific discussions and everyday uses of temperature.
  • Scales are created based on fixed reference points, like the freezing and boiling points of substances.
  • Conversions between scales can be managed through formulas, often deriving from linear relationships.
  • New scales, like the one involving ethanol's melting and boiling points, can help illustrate temperature concepts or adapt readings for specific contexts.
Linear Equations
Linear equations are mathematical expressions that describe straight lines. In temperature conversions, linear equations define the relationship between two temperature scales. Such relationships are crucial when designing a conversion formula, as seen in our task.

The general form of a linear equation is given by \[ S = kC + b \]where:
  • \(S\) is the value in the new temperature scale.
  • \(C\) is the temperature in Celsius.
  • \(k\) is the slope, showing how many units \(S\) changes with a unit change in \(C\).
  • \(b\) is the intercept, the starting point on the \(S\)-axis when \(C = 0\).
This equation helps us find the equivalent temperature on the new scale for any given Celsius temperature. The slope indicates the direct relationship between the scales, while the intercept adjusts the starting point to accommodate the different reference points.
Ethanol Melting and Boiling Points
In this problem, we use ethanol to develop a new temperature scale. Ethanol has distinct melting and boiling points at \(-117.3^{\circ} \mathrm{C}\) and \(78.3^{\circ} \mathrm{C}\) respectively. On the new scale created, these become \(0^{\circ} \mathrm{S}\) and \(100^{\circ} \mathrm{S}\).

Understanding the melting and boiling points of ethanol is essential since it helps define the endpoints of the new temperature scale. These benchmarks establish clear reference points for mapping Celsius to the new scale \(\mathrm{S}\).
  • Freezing point of ethanol: \(-117.3^{\circ} \mathrm{C}\) = \(0^{\circ} \mathrm{S}\).
  • Boiling point of ethanol: \(78.3^{\circ} \mathrm{C}\) = \(100^{\circ} \mathrm{S}\).
  • Key to creating the scale: these reference points set the linear span of the scale.
This understanding makes it possible to create a linear equation to convert Celsius to \(\mathrm{S}\), providing a practical application and experimentation method to grasp temperature conversion principles.

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Most popular questions from this chapter

Dinosaurs dominated life on Earth for millions of years and then disappeared very suddenly. In the experimentation and data-collecting stage, paleontologists studied fossils and skeletons found in rocks in various layers of Earth's crust. Their findings enabled them to map out which species existed on Earth during specific geologic periods. They also revealed no dinosaur skeletons in rocks formed immediately after the Cretaceous period, which dates back some 65 million years. It is therefore assumed that the dinosaurs became extinct about 65 million years ago. Among the many hypotheses put forward to account for their disappearance were disruptions of the food chain and a dramatic change in climate caused by violent volcanic eruptions. However, there was no convincing evidence for any one hypothesis until 1977\. It was then that a group of paleontologists working in Italy obtained some very puzzling data at a site near Gubbio. The chemical analysis of a layer of clay deposited above sediments formed during the Cretaceous period (and therefore a layer that records events occurring after the Cretaceous period) showed a surprisingly high content of the element iridium. Iridium is very rare in Earth's crust but is comparatively abundant in asteroids. This investigation led to the hypothesis that the extinction of dinosaurs occurred as follows. To account for the quantity of iridium found, scientists suggested that a large asteroid several miles in diameter hit Earth about the time the dinosaurs disappeared. The impact of the asteroid on Earth's surface must have been so tremendous that it literally vaporized a large quantity of surrounding rocks, soils, and other objects. The resulting dust and debris floated through the air and blocked the sunlight for months or perhaps years. Without ample sunlight most plants could not grow, and the fossil record confirms that many types of plants did indeed die out at this time. Consequently, of course, many plant-eating animals gradually perished, and then, in turn, meat-eating animals began to starve. Limitation of food sources obviously affects large animals needing great amounts of food more quickly and more severely than small animals. Therefore, the huge dinosaurs vanished because of lack of food. (a) How does the study of dinosaur extinction illustrate the scientific method? (b) Suggest two ways to test the hypothesis. (c) In your opinion, is it justifiable to refer to the asteroid explanation as the theory of dinosaur extinction? (d) Available evidence suggests that about 20 percent of the asteroid's mass turned to dust and spread uniformly over Earth after eventually settling out of the upper atmosphere. This dust amounted to about \(0.02 \mathrm{~g} / \mathrm{cm}^{2}\) of Earth's surface. The asteroid very likely had a density of about \(2 \mathrm{~g} / \mathrm{cm}^{3}\). Calculate the mass (in kilograms and tons) of the asteroid and its radius in meters, assuming that it was a sphere. (The area of Earth is \(5.1 \times 10^{14} \mathrm{~m}^{2} ; 1 \mathrm{lb}=453.6 \mathrm{~g} .\) ) (Source: Consider a Spherical Cow-A Course in Environmental Problem Solving by J. Harte, University Science Books, Mill Valley, CA, 1988 . Used with permission.

At what temperature does the numerical reading on a Celsius thermometer equal that on a Fahrenheit thermometer?

Does each of these describe a physical change or a chemical change? (a) The helium gas inside a balloon tends to leak out after a few hours. (b) A flashlight beam slowly gets dimmer and finally goes out. (c) Frozen orange juice is reconstituted by adding water to it. (d) The growth of plants depends on the sun's energy in a process called photosynthesis. (e) A spoonful of table salt dissolves in a bowl of soup.

This procedure was carried out to determine the volume of a flask. The flask was weighed dry and then filled with water. If the masses of the empty flask and the filled flask were \(56.12 \mathrm{~g}\) and \(87.39 \mathrm{~g}\), respectively, and the density of water is \(0.9976 \mathrm{~g} / \mathrm{cm}^{3},\) calculate the volume of the flask in cubic centimeters.

Write the equations for converting degrees Celsius to degrees Fahrenheit and degrees Fahrenheit to degrees Celsius.

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