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Chapter 12: Problem 9

ssm On the Rankine temperature scale, which is sometimes used in engineering applications, the ice point is at \(491.67^{\circ}{R}\) and the steam point is at \(671.67^{\circ}{R}\). Determine a relationship (analogous to Equation 12.1) between the Rankine and Fahrenheit temperature scales.

Short Answer

Expert verified
The conversion formula is \( T_{R} = T_{F} + 459.67 \).

Step by step solution

01

Understand the Problem

Equation 12.1 in many contexts refers to the linear relationship between two temperature scales. Here, we aim to find a linear equation relating Rankine (R) to Fahrenheit (F) much like how Kelvin relates to Celsius. We are given the ice point and steam point temperatures in Rankine, commonly known reference points in temperature conversion studies.
02

Determine Base Reference Points

The ice point (the freezing point of water) is given as \(491.67^{\circ}{R}\) and corresponds to \(32^{\circ}{F}\). The steam point (boiling point of water) is at \(671.67^{\circ}{R}\) and corresponds to \(212^{\circ}{F}\). These points will be used to understand the scale differences and start formulating our equation.
03

Calculate the Conversion Ratio

Calculate the difference in each scale between the steam point and the ice point:- For Rankine: \(671.67^{\circ}{R} - 491.67^{\circ}{R} = 180^{\circ}{R}\)- For Fahrenheit: \(212^{\circ}{F} - 32^{\circ}{F} = 180^{\circ}{F}\)This tells us that the size of one degree in Rankine is the same as one degree in Fahrenheit.
04

Establish a Formula

Since both scales move one unit per degree in both Rankine and Fahrenheit, the formula relating Rankine and Fahrenheit is linear. The formula is:\[ T_{R} = T_{F} + 459.67 \]or inversely,\[ T_{F} = T_{R} - 459.67 \]where \(T_{R}\) is the temperature in Rankine, and \(T_{F}\) is the temperature in Fahrenheit.
05

Verify the Formula

Verify the relationship with the given ice and steam points:- For the ice point, \(T_{F} = 491.67 - 459.67 = 32^{\circ}{F}\)- For the steam point, \(T_{F} = 671.67 - 459.67 = 212^{\circ}{F}\)The calculation confirms our linear formula holds true for these points.

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Key Concepts

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

Temperature Conversion
Temperature conversion is a method used to translate a value from one temperature scale to another. Understanding temperature conversion is essential, especially when dealing with different scientific and industrial settings. Two temperature scales that interact in this context are Rankine (掳R) and Fahrenheit (掳F).

The Rankine scale, like Kelvin, starts from absolute zero but uses the Fahrenheit increment size. Conversely, the Fahrenheit scale has a defined zero point related to the freezing and boiling points of water. For instance, the freezing point of water is at 32掳F and 491.67掳R, while the boiling point is at 212掳F and 671.67掳R.
  • The equation for converting Rankine to Fahrenheit is: \[ T_{R} = T_{F} + 459.67 \]
  • And inversely, converting Fahrenheit to Rankine is: \[ T_{F} = T_{R} - 459.67 \]
This formula allows anyone to switch between these temperature measures with ease, broadening the application in various engineering fields.
Linear Temperature Relationship
A linear temperature relationship means that as the temperature in one scale changes, the temperature in another scale changes proportionally. This relationship is represented mathematically by a straight-line equation. In the case of the Rankine and Fahrenheit scales:
  • The temperature change in one scale matches the change in the other.
  • This proportional change makes the relationship between these scales linear.
For Rankine and Fahrenheit, each unit increase in Rankine corresponds to a unit increase in Fahrenheit, as proven by the differences between the freezing and boiling points.

These points both show a 180-degree difference in their respective scales. It confirms that the scales change at the same rate, maintaining a linear correspondence across the range. This simplicity in their change rate allows simplistic conversion calculations.
Fahrenheit Scale
The Fahrenheit scale is a temperature scale that is primarily used in the United States and some Caribbean countries. It was developed by Daniel Gabriel Fahrenheit, where he referenced the freezing point of water at 32掳F and the boiling point at 212掳F. This scale is less commonly used in scientific processes, which often prefer Celsius or Kelvin.

In the context of temperature conversions with Rankine:
  • The Rankine scale shares the same degree magnitude as Fahrenheit.
  • Absolute zero on the Fahrenheit scale occurs at -459.67掳F, which is the starting point for the Rankine scale.
These aspects make conversions straightforward between these two scales due to their shared Unit Size. Understanding the Fahrenheit scale is crucial when dealing with older systems or certain regional applications where it is still the main temperature measure.

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Most popular questions from this chapter

A spherical brass shell has an interior volume of \(1.60 \times 10^{-3} {m}^{3}\) Within this interior volume is a solid steel ball that has a volume of \(0.70 \times 10^{-3} {m}^{3}\) . The space between the steel ball and the inner surface of the brass shell is filled completely with mercury. A small hole is drilled through the brass, and the temperature of the arrangement is increased by 12 \({C}^{\circ}\). What is the volume of the mercury that spills out of the hole?

Two bars of identical mass are at \(25^{\circ} {C}\) . One is made from glass and the other from another substance. The specific heat capacity of glass is 840 \({J} /({kg} \cdot {C}^{\circ})\) . When identical amounts of heat are supplied to each, the glass bar reaches a temperature of \(88^{\circ} {C}\) , while the other bar reaches 250.0 'C. What is the specific heat capacity of the other substance?

On the moon the surface temperature ranges from 375 K during the day to \(1.00 \times 10^{2} {K}\) at night. What are these temperatures on the (a) Celsius and (b) Fahrenheit scales?

An aluminum can is filled to the brim with a liquid. The can and the liquid are heated so their temperatures change by the same amount. The can's initial volume at \(5^{\circ} {C}\) is \(3.5 \times 10^{-4} {m}^{3} .\) The coefficient of volume expansion for aluminum is \(69 \times 10^{-6}({C}^{9})^{-1}\) When the can and the liquid are heated to \(78^{\circ} {C}, 3.6 \times 10^{-6} {m}^{3}\) of liquid spills over. What is the coefficient of volume expansion of the liquid?

You are sick, and your temperature is 312.0 kelvins. Convert this temperature to the Fahrenheit scale.
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