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The Rankine temperature scale (abbreviated \(^{\circ} \mathrm{R}\) ) uses the same size degrees as Fahrenheit, but measured up from absolute zero like kelvin (so Rankine is to Fahrenheit as kelvin is to Celsius). Find the conversion formula between Rankine and Fahrenheit, and also between Rankine and kelvin. What is room temperature on the Rankine scale?

Short Answer

Expert verified
Rankine to Fahrenheit: \( T_{F} = T_{R} - 459.67 \). Rankine to Kelvin: \( T_{K} = T_{R} \times \frac{5}{9} \). Room temperature: 527.67 °R.

Step by step solution

01

Understanding Absolute Zero

Absolute zero is the lowest possible temperature, which is 0 K (kelvin) or 0 °R (Rankine). This serves as the starting point for both the kelvin and Rankine scales.
02

Relation to Fahrenheit Scale

Since the Rankine scale is simply the Fahrenheit scale shifted down to absolute zero, the two scales have the same incremental size. A temperature in °R can be converted to °F by subtracting 459.67, which is the conversion for absolute zero in Fahrenheit.
03

Conversion Formula from Rankine to Fahrenheit

To convert a temperature from Rankine to Fahrenheit, use the formula: \( T_{\mathrm{F}} = T_{\mathrm{R}} - 459.67 \). This accounts for the shift in the baseline reference point, as absolute zero is 459.67 °F.
04

Relation to Kelvin Scale

Both the Rankine and Kelvin scales start from absolute zero but use different incremental degrees. The Kelvin scale is based on Celsius, hence, the conversion from Kelvin to Rankine is direct by a factor: \( T_{\mathrm{R}} = T_{\mathrm{K}} \times \frac{9}{5} \).
05

Conversion Formula from Rankine to Kelvin

To convert from Rankine to Kelvin, use \( T_{\mathrm{K}} = T_{\mathrm{R}} \times \frac{5}{9} \). This equation aligns the Rankine scale's size with the Kelvin scale's.
06

Converting Room Temperature to Rankine

Room temperature is typically around 20-22°C, equivalent to 293.15 K. Using the conversion \( T_{\mathrm{R}} = T_{\mathrm{K}} \times \frac{9}{5} \), we find the room temperature in Rankine: \( T_{\mathrm{R}} = 293.15 \times \frac{9}{5} = 527.67 \) °R.

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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 all about changing a temperature from one scale to another. It's like turning kilometers into miles when you're talking about a journey. Each temperature scale has its way of marking degrees, and they start their measure at different reference points.

When you're converting temperatures, you're moving numbers from one system to another. In this case, to change Rankine to Fahrenheit, you subtract because Fahrenheit counts up from a warmer start point compared to Rankine, which starts at the coldest point possible, absolute zero. Conversely, converting between Rankine and Kelvin involves a simple multiplication or division because both start at absolute zero but are scaled differently.

Remembering these relationships helps a lot:
  • Fahrenheit and Rankine both share degree sizes but differ in their zero point.
  • Kelvin is Celsius with a shift in baseline to match absolute zero, similar to Rankine's relation to Fahrenheit.
Recognizing these connections makes it second nature to hop between scales.
Fahrenheit Scale
The Fahrenheit scale measures temperature with degrees that are smaller than those in the Celsius scale. It's commonly used in the United States for most everyday temperature readings, like weather, cooking, and human body temperature.

The baseline for Fahrenheit is set with ice melting (32 °F) and water boiling (212 °F) under sea-level conditions. These points were chosen because they represent physical phenomena that are easy to reproduce and check.

However, when we talk about the absolute zero in Fahrenheit, that's -459.67 °F. This is the point where no more heat energy remains to be extracted from a system. It’s a starting mark for the Rankine scale, making it quite a shift from the familiar points of water freezing and boiling. Adjust consequently when shifting from Rankine to Fahrenheit, as you subtract 459.67.
Kelvin Scale
The Kelvin scale is the scientist's go-to for measuring temperature. It's neat because it starts with absolute zero, the coldest possible temperature where everything theoretically stops moving. It bypasses negative numbers, which is often tidy for equations and scientific endeavors.

Kelvin uses the same degree size as Celsius, and the conversion is straightforward: simply add 273.15 to a Celsius temperature to get it in Kelvin. With no negative numbers, calculations involving energy and heat become easier and more intuitive.
Alongside Rankine, Kelvin is vital for sciences combining heat, light, or radiation because of its direct zero-start from absolute zero. To switch from Rankine, multiply or divide by 9/5 because their zero point, absolute zero, is common, but the degree scale varies.
Absolute Zero
Absolute zero is the theoretical point where all molecular movement ceases; it's the lowest energy state a system can achieve. In Kelvin, it's precisely 0 K, and in Rankine, it’s the same at 0 °R, underscoring the concept's fundamental nature across scales.

Achieving absolute zero is impossible practically due to quantum mechanics’ principles, which dictate that some small activity always persists. However, laboratories strive to come close to it for various experiments that explore the limits of physical laws.

The point of absolute zero is pivotal to many scientific fields, from quantum computing to superconductivity studies. Understanding it helps explain why temperatures are measured from such a uniquely defined zero point in scientific scales like Kelvin and Rankine.

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Most popular questions from this chapter

Suppose you have a gas containing hydrogen molecules and oxygen molecules, in thermal equilibrium. Which molecules are moving faster, on average? By what factor?

Consider a uniform rod of material whose temperature varies only along its length, in the \(x\) direction. By considering the heat flowing from both directions into a small segment of length \(\Delta x,\) derive the heat equation, $$ \frac{\partial T}{\partial t}=K \frac{\partial^{2} T}{\partial x^{2}} $$ where \(K=k_{t} / c \rho, c\) is the specific heat of the material, and \(\rho\) is its density. (Assume that the only motion of energy is heat conduction within the rod; no energy enters or leaves along the sides.) Assuming that \(K\) is independent of temperature, show that a solution of the heat equation is $$ T(x, t)=T_{0}+\frac{A}{\sqrt{t}} e^{-x^{2} / 4 K t} $$ where \(T_{0}\) is a constant background temperature and \(A\) is any constant. Sketch (or use a computer to plot) this solution as a function of \(x,\) for several values of \(t\) Interpret this solution physically, and discuss in some detail how energy spreads through the rod as time passes.

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