Question

Another temperature scale is the Rankine scale. It represents an absolute temperature scale similar to the Kelvin scale, with a common absolute zero (i.e., \(0.0 \mathrm{~K}=0.0{ }^{\circ} \mathrm{R}\) ). However, whereas a change of \(1.0 \mathrm{~K}\) is the same as a change of \(1.0^{\circ} \mathrm{C}\), a change of \(1.0{ }^{\circ} \mathrm{R}\) is the same as \(1.0{ }^{\circ} \mathrm{F}\). Absolute zero on the Rankine scale equals \(-459.67{ }^{\circ} \mathrm{F}\). Water freezes at \(32{ }^{\circ} \mathrm{F}\) (or \(\left.0.0{ }^{\circ} \mathrm{C}\right)\) and boils at \(212{ }^{\circ} \mathrm{F}\) \(\left(100.0^{\circ} \mathrm{C}\right)\). Convert these temperatures to their equivalent temperatures on the Rankine scale.

Short answer

Freezing point: \(491.67 ^\circ R\), Boiling point: \(671.67 ^\circ R\).

Step-by-step solution

Understanding Rankine Conversion

To convert Fahrenheit to Rankine, use the formula \[ R = F + 459.67 \]. This accounts for the difference in zero points between the two scales, where Rankine's absolute zero is \(-459.67 ^\circ F\).

Freezing Point Conversion

Water freezes at \(32 ^\circ F\). To find the Rankine equivalent, calculate:\[R = 32 + 459.67 = 491.67 ^\circ R\]This gives the freezing point in Rankine.

Boiling Point Conversion

Water boils at \(212 ^\circ F\). Use the conversion formula:\[R = 212 + 459.67 = 671.67 ^\circ R\]This gives the boiling point in Rankine.

Key concepts

Temperature Conversion

Temperature conversion is the process of changing a temperature reading from one scale to another.
There are several temperature scales, each used in different parts of the world or for specific scientific purposes.

Common temperature scales include:

• Celsius (°C) - commonly used worldwide in daily life and weather reports.
• Fahrenheit (°F) - mainly used in the United States.
• Kelvin (K) - the standard for scientific calculations, often used in physics and chemistry.
• Rankine (°R) - primarily used in thermodynamics and some engineering fields.

Temperature conversion equations are derived based on the relationships between the scales.
For instance, converting from Celsius to Fahrenheit involves the formula \( F = \frac{9}{5}C + 32 \).
Conversion between Fahrenheit and Rankine is simpler due to the direct relationship between the scales.
Knowing these formulas helps in understanding how each temperature scale can be compared or converted to another.

Absolute Temperature Scale

The absolute temperature scale is fundamental in understanding thermodynamics and the laws of physics.
It is a scale where temperatures are measured from absolute zero, the theoretical point where particle motion ceases.

The two main absolute temperature scales are:

• Kelvin (K) - Widely used in scientific studies. It aligns with Celsius but starts at absolute zero.
• Rankine (°R) - Similar to Kelvin, but ties into the Fahrenheit system, maintaining its unit size.

Understanding absolute temperature scales is crucial for fields such as thermodynamics, where precise calculations related to energy and heat flow are needed.
These scales provide a consistent basis for comparing temperature readings across different systems and experiments.
Scientists and engineers rely heavily on the concept of absolute zero to gauge thermal behavior accurately.

Fahrenheit to Rankine Conversion

Converting Fahrenheit to Rankine is a straightforward process due to the direct mathematical relationship between the two scales.
The Rankine scale is an extension of the Fahrenheit scale, adding 459.67 to account for absolute zero in Fahrenheit.

Conversion Formula:- The formula for converting Fahrenheit to Rankine is: \[ R = F + 459.67 \] Here's how it works:
- For a temperature at the freezing point of water, 32 °F, the conversion to Rankine is \( R = 32 + 459.67 = 491.67 ^\circ R \).
- At the boiling point of water, 212 °F, it converts to \( R = 212 + 459.67 = 671.67 ^\circ R \).
This conversion is useful in various scientific and engineering applications where Rankine is preferred.
The ease of conversion makes it a practical choice for situations where Fahrenheit measurements are more common.
By understanding this simple formula, students can easily convert between these temperature scales as needed.