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Chapter 9: Problem 33

Absolute zero on a temperature scale called the Rankine scale is \(T_{R}=0^{\circ} \mathrm{R},\) and the scale's unit is the same size as the Fahrenheit degree. a. Write a formula that relates the Rankine scale to the Fahrenheit scale. b. Write a formula that relates the Rankine scale to the Kelvin scale.

Short Answer

Expert verified
The formula that relates Rankine scale to Fahrenheit scale is: \(T_{R} = T_{F} + 459.67\). The formula that relates Rankine scale to Kelvin scale is: \(T_{R} = 1.8 \cdot T_{K}\).

Step by step solution

01

Fahrenheit to Rankine

The Rankine and Fahrenheit scales both have the same unit size, and they both define absolute zero as 0. So to convert from Fahrenheit to Rankine we simply add 459.67, which is the value of absolute zero in Fahrenheit. The formula is: \(T_{R} = T_{F} + 459.67\) where \(T_{R}\) is temperature in Rankine scale and \(T_{F}\) is temperature in Fahrenheit scale.
02

Kelvin to Rankine

The Rankine scale is related to the Kelvin scale by a factor of 1.8, since one Kelvin is the equivalent of 1.8 Rankine degrees. This is a linear conversion. Therefore, we multiply the temperature in Kelvin by 1.8 to get the temperature in the Rankine scale. The formula is: \(T_{R} = 1.8 \cdot T_{K}\) where \(T_{R}\) is temperature in Rankine scale and \(T_{K}\) is temperature in Kelvin scale.

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Key Concepts

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

Rankine Scale
The Rankine scale is a temperature scale similar to Kelvin, but it uses the Fahrenheit degree as its unit increment.
Absolute zero on the Rankine scale is at 0 掳R, just like absolute zero in Kelvin is 0 K. The Rankine scale is not as widely used as Celsius or Kelvin, but it can be particularly useful in engineering systems where Fahrenheit is used, especially in the United States.
To convert Fahrenheit to Rankine, you simply add 459.67 to the Fahrenheit temperature. This is because absolute zero on the Fahrenheit scale is -459.67 掳F. The conversion formula is straightforward:
  • The formula: \[ T_R = T_F + 459.67 \] where \( T_R \) represents the temperature in degrees Rankine, and \( T_F \) is the temperature in degrees Fahrenheit.
Fahrenheit Scale
The Fahrenheit scale is a commonly used temperature scale, particularly in the United States. Water freezes at 32 掳F and boils at 212 掳F under standard atmospheric conditions. Unlike the Celsius and Kelvin scales, the degree intervals in Fahrenheit are smaller, which allows for more precise readings of temperature changes.
On the Fahrenheit scale, absolute zero is set at -459.67 掳F. It aligns with the Rankine scale to help convert measurements. As mentioned earlier, to shift to the Rankine scale you just add 459.67. This connection highlights the relationship between Fahrenheit and temperature scales that use absolute zero as a reference point.
Engineering often relies on this scale for its precision in common temperature ranges and its relation to the Rankine scale used for some thermodynamic calculations.
Kelvin Scale
The Kelvin scale is a metric temperature scale that is frequently used in scientific and engineering disciplines because it begins at absolute zero, the point where all thermal motion of atoms theoretically ceases.
The unit of measurement, Kelvin, is the same size as a degree Celsius but starts at absolute zero (0 K), making conversions between the Celsius and Kelvin scales straightforward:
  • 1 K increment is the same as a 1 掳C increment, but 0 K is approximately -273.15 掳C.
The Kelvin scale aligns well with other temperature scales through linear conversions. To convert Kelvin to Rankine, you multiply the temperature in Kelvin by 1.8. This reflects the size difference between the Celsius/ Kelvin and the Fahrenheit/ Rankine units. The conversion formula is:
  • The formula: \[ T_R = 1.8 \times T_K \] where \( T_R \) is the temperature in Rankine, and \( T_K \) is the temperature in Kelvin.
This formula is crucial for calculations that need to incorporate temperature scale differences between international standards and U.S. conventions.

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