Problem 10 - University Physics with Modern Physics

Question

Like the Kelvin scale, the Rankine scale is an absolute temperature scale: Absolute zero is zero degrees Rankine (0\(^\circ\)R). However, the units of this scale are the same size as those of the Fahrenheit scale rather than the Celsius scale. What is the numerical value of the triple-point temperature of water on the Rankine scale?

Step-by-Step Solution

Answer

The triple-point temperature of water on the Rankine scale is 491.688°R.

Step 1: Understand the Triple-Point and Conversion

The triple-point temperature of water is a fundamental fixed point which is defined as 273.16 Kelvin (K). Our task is to express this temperature in the Rankine scale. We need the conversion relationship between Kelvin and Rankine.

Step 2: Conversion Formula from Kelvin to Rankine

The conversion from Kelvin to Rankine is done using the formula: \[ T_{R} = T_{K} \times \frac{9}{5} \]where \( T_{R} \) is the temperature in Rankine and \( T_{K} \) is the temperature in Kelvin. It is important to understand that 0 Rankine is absolute zero, just like 0 Kelvin.

Step 3: Plug in the Triple-Point Temperature of Water

Now we substitute 273.16 K into the conversion formula:\[ T_{R} = 273.16 \times \frac{9}{5} \]

Step 4: Perform the Calculation

Let's calculate:\[ T_{R} = 273.16 \times \frac{9}{5} = 491.688 \]Thus, the triple-point temperature of water expressed on the Rankine scale is 491.688°R.

Key Concepts

Absolute Temperature Scale

The Rankine and Kelvin scales are both examples of absolute temperature scales. An absolute temperature scale is one where the lowest possible temperature is absolute zero. Absolute zero represents the point at which the molecules of a substance have minimal possible motion, which physically cannot be surpassed. In essence, absolute zero marks the boundary of thermal energy existence, influencing how these scales are developed.
This type of scale measures temperature with a universal starting point, unlike the Celsius or Fahrenheit scales, which are based on arbitrary points, like the freezing point of water. On absolute scales like Kelvin and Rankine, absolute zero is defined as 0 K and 0 °R, ensuring a consistent baseline for scientific measurements. Using absolute temperature scales is essential in scientific fields such as thermodynamics, where temperature values need to avoid negative numbers that can complicate theoretical calculations or simulations.

Kelvin to Rankine Conversion

Converting between Kelvin and Rankine is a straightforward task, as both scales are absolute and linear but differ in their degree sizes. This difference arises due to their historical origins. Kelvin is based on the metric Celsius scale, while Rankine is based on the imperial Fahrenheit scale.
To convert Kelvin to Rankine, multiply the temperature in Kelvin by the fraction \( \frac{9}{5} \).

• This ratio reflects the relationship between Celsius and Fahrenheit, as each step in the Celsius scale (used in Kelvin) is \( \frac{5}{9} \) of the step in the Fahrenheit scale (used in Rankine).
• Because they start at absolute zero, the conversion formula directly scales temperatures without needing to adjust with an additional offset, like adding 273.15 for Celsius to Kelvin or 32 for Celsius to Fahrenheit.

Recognizing this simplicity makes conversions easy and quick, an advantage in scientific processes demanding rapid analysis.

Triple-Point Temperature of Water

One of the critical benchmarks in thermodynamics is the triple-point temperature of water. By definition, this is the specific condition where water coexists in all three states: solid, liquid, and gas, establishing a fundamental temperature context.
In the Kelvin scale, this point is precisely set at 273.16 K, which is used as a fixed reference in both Kelvin and Rankine due to its reliability across different situations. This consistency allows it to serve as a pivotal reference for calibrating thermodynamic experiments or developing scientific instruments.
When calculating this temperature on the Rankine scale, the conversion formula \(T_{R} = T_{K} \times \frac{9}{5}\) is employed, leading to a numerical value of 491.688°R. This conversion not only underscores the unity between the absolute temperature scales but also the importance of having standardized reference points like the triple-point of water, which help to unify scientific research for consistent outcomes.