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Chapter 1: Problem 102

In 1701 , the Danish astronomer Ole Romer proposed a temperature scale with two fixed points, freezing water at 7.5 degrees, and boiling water at 60.0 degrees. What is the boiling point of oxygen, \(90.2 \mathrm{K}\), on the Romer scale?

Short Answer

Expert verified
The boiling point of oxygen in the Romer scale is approximately \(-58.6\) degrees.

Step by step solution

01

Determine the conversion coefficients

Let \(K\) be a temperature in Kelvin and \(R\) the corresponding temperature in Romer. We assume that there is a linear relationship: \[ R = aK + b \] We have two reference points: 1. Freezing point of water: \((K=273.15, R=7.5)\) 2. Boiling point of water: \((K=373.15, R=60)\) Plugging these values into our formula, we get the following system of equations: \[ \begin{cases} 7.5 = a(273.15) + b \\ 60.0 = a(373.15) + b \end{cases} \] Now, we'll solve this system of equations for the coefficients \(a\) and \(b\).
02

Solve the system of equations for a and b

We can subtract the first equation from the second equation to eliminate \(b\): \[ 52.5 = a(373.15 - 273.15) \] Now, let's solve for \(a\): \[ a = \frac{52.5}{373.15 - 273.15} = \frac{52.5}{100} = 0.525 \] Now, to find \(b\), we substitute this value of \(a\) into either of the equations, for example, the first equation: \[ 7.5 = 0.525(273.15) + b \] \[ b = 7.5 - 0.525 \times 273.15 \approx -135.9 \] So the linear conversion formula between Kelvin and Romer is: \[ R = 0.525K - 135.9 \]
03

Find the boiling point of oxygen in the Romer scale

Now that we have a conversion formula between Kelvin and Romer, we can find the boiling point of oxygen in the Romer scale by substituting \(K=90.2\) (the boiling point of oxygen in Kelvin) into the formula: \[ R_{oxygen} = 0.525(90.2) - 135.9 \] \[ R_{oxygen} \approx -58.595 \] So the boiling point of oxygen in the Romer scale is approximately \(-58.6\) degrees.

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Key Concepts

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

Romer Scale
The Romer scale is a historical temperature scale introduced by the Danish astronomer Ole Romer in 1701. It is one of the less commonly known scales but holds significance in the development of temperature measurement. Romer set his scale based on two fixed reference points.
The freezing point of water is 7.5 degrees on the Romer scale, while the boiling point is set at 60 degrees. This uniqueness lies in the non-zero-based division, unlike the more modern Celsius scale, which starts at zero.
In modern terms, the Romer scale is not widely used, as scales like Celsius, Fahrenheit, and Kelvin provide more standardized measurements. However, understanding the Romer scale helps appreciate the evolution of scientific measurements.
Kelvin to Romer conversion
Converting temperatures from Kelvin to the Romer scale requires understanding the relationship between these two scales. This relationship is based on the assumption of a linear conversion equation.
The general form of converting Kelvin ( K ) to Romer ( R ) is R = aK + b , where a and b are constants found through reference points. By using the freezing and boiling points of water, equations are formed and solved to determine these values.
Thus, through calculation, the conversion formula is derived as R = 0.525K - 135.9 , which allows us to convert any Kelvin temperature into the Romer scale effectively. This method allows for an accurate transformation when comparing temperatures across different scales.
Linear equation solution
In the context of temperature conversion, solving a system of linear equations is a crucial mathematical step. When tasked with finding the relationship between Kelvin and Romer, we assume a linear relationship: R = aK + b .
By using two known points (water's freezing and boiling points in both scales), two equations are formed and allow solving for a and b using simple algebraic steps.
Linear equations express proportional relationships and solving them involves steps like substitution and elimination. Solving them accurately is essential for constructing accurate conversion formulas, such as finding a = 0.525 and b = -135.9 in our exercise, which achieves a functional Kelvin to Romer conversion formula.
Ole Romer
Ole Romer, born in Denmark in 1644, was a pioneering astronomer known for several scientific advancements, particularly in measuring the speed of light. Yet, he also contributed to temperature measurements by developing the Romer scale.
His work symbolizes the quest for precision in scientific observations during a time when scientific instruments were becoming crucial for exploration and discovery.
While his temperature scale isn鈥檛 commonly used today, Romer鈥檚 efforts laid the groundwork for future improvements in temperature measurement, demonstrating the importance of his work in the history of science.

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Most popular questions from this chapter

Even when shut down after a period of normal use, a large commercial nuclear reactor transfers thermal energy at the rate of \(150 \mathrm{MW}\) by the radioactive decay of fission products. This heat transfer causes a rapid increase in temperature if the cooling system fails (1 watt \(=1\) joule/second or \(1 \mathrm{W}=1 \mathrm{J} / \mathrm{s}\) and \(1 \mathrm{MW}=1 \text { megawatt }) . \quad\) (a) Calculate the rate of temperature increase in degrees Celsius per second ( \(^{\circ} \mathrm{C} / \mathrm{s}\) ) if the mass of the reactor core is \(1.60 \times 10^{5} \mathrm{kg}\) and it has an average specific heat of \(0.3349 \mathrm{kJ} / \mathrm{kg} \cdot^{\circ} \mathrm{C}\). (b) How long would it take to obtain a temperature increase of \(2000^{\circ} \mathrm{C},\) which could cause some metals holding the radioactive materials to melt? (The initial rate of temperature increase would be greater than that calculated here because the heat transfer is concentrated in a smaller mass. Later, however, the temperature increase would slow down because the \(500,000-\mathrm{kg}\) steel containment vessel would also begin to heat up.)

Give an example in which \(A\) has some kind of nonthermal equilibrium relationship with \(B\), and \(B\) has the same relationship with \(C,\) but \(A\) does not have that relationship with \(C\).

You pour coffee into an unlidded cup, intending to drink it 5 minutes later. You can add cream when you pour the cup or right before you drink it. (The cream is at the same temperature either way. Assume that the cream and coffee come into thermal equilibrium with each other very quickly.) Which way will give you hotter coffee? What feature of this question is different from the previous one?

How is heat transfer related to temperature?

In a physics classroom demonstration, an instructor inflates a balloon by mouth and then cools it in liquid nitrogen. When cold, the shrunken balloon has a small amount of light blue liquid in it, as well as some snow-like crystals. As it warms up, the liquid boils, and part of the crystals sublime, with some crystals lingering for a while and then producing a liquid. Identify the blue liquid and the two solids in the cold balloon. Justify your identifications using data from Table 1.4.
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