Problem 14 - University Physics with Modern Physics

Question

Like the Kelvin scale, the Rankine scale is an absolute temperature scale: Absolute zero is zero degrees Rankine \(\left(0^{\circ} \mathrm{R}\right)\) . 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 is 491.688 °R on the Rankine scale.

Step 1: Understanding the Triple-point

The triple-point of water is the temperature at which water can exist in equilibrium as a solid, liquid, and gas. On the Kelvin scale, this temperature is precisely 273.16 K.

Step 2: Converting Kelvin to Rankine

Rankine (°R) is related to Kelvin (K) through the conversion factor: \[ 1 \, \text{K} = \frac{9}{5} \, \text{°R} \] This means to convert from Kelvin to Rankine, we multiply the Kelvin temperature by \( \frac{9}{5} \).

Step 3: Applying the Conversion

Given the triple-point temperature of water is 273.16 K, we apply the conversion:\[ \text{Triple-point in Rankine} = 273.16 \times \frac{9}{5} \]

Step 4: Calculating the Result

Calculate the product:\[ 273.16 \times \frac{9}{5} = 491.688 \, \text{°R} \]So, the triple-point temperature of water in the Rankine scale is approximately 491.688 °R.

Key Concepts

Kelvin Scale

The Kelvin scale is a thermodynamic temperature scale used widely in scientific contexts, particularly because of its absolute nature. Named after the scientist Lord Kelvin, this scale is designed so that absolute zero—the lowest possible temperature where molecular motion ceases—is defined as 0 K (zero Kelvin). Unlike the Celsius scale, which begins at 0°C for the freezing point of water and 100°C for its boiling point, the Kelvin scale starts at absolute zero and moves upward. A unique feature of Kelvin is that its increments are equivalent to those on the Celsius scale. This means that an increase of 1 Kelvin is identical to an increase of 1°C. This direct relationship facilitates easy conversion between Kelvin and Celsius, where the conversion formula is:

• Temperature in Kelvin = Temperature in Celsius + 273.15.

This relationship makes the Kelvin scale very handy for scientifically calculating absolute temperatures.

Temperature Conversion

Temperature conversion is a fundamental process in comparing temperature measurements across different scales. Celsius, Fahrenheit, Kelvin, and Rankine are all temperature scales that often require conversion for proper understanding in different contexts. Knowing how to convert between these scales is crucial in science and engineering disciplines.
For instance, when converting Kelvin to Rankine, understanding their relationship is essential. The conversion involves a simple multiplicative factor:

• \(1 \, \text{K} = \frac{9}{5} \, \text{°R}\).

To convert a given temperature in Kelvin (K) to Rankine (°R), multiply the Kelvin temperature by \(\frac{9}{5}\). This conversion is based on aligning the two scales without an offset, unlike the Celsius-to-Fahrenheit conversion, which involves both multiplication and addition given by:

• \(\text{Temperature in Fahrenheit} = \text{Temperature in Celsius} \times \frac{9}{5} + 32.\)

Effective temperature conversion is crucial for consistency in scientific measurement and communication.

Triple-point of Water

The triple-point of water is a significant concept in thermodynamics and physical chemistry. It refers to the unique set of conditions at which water can coexist in equilibrium in all three states: solid (ice), liquid (water), and gas (vapor). At this precise point, which occurs at a temperature of 273.16 K and a pressure of 611.657 pascals, the states are in perfect balance.
The triple-point serves as a fixed point for the definition of temperature scales. For example, it is used to define the Kelvin scale: 1 Kelvin unit is \(\frac{1}{273.16}\) of the thermodynamic temperature of the triple-point of water. This characteristic makes the triple-point a critical reference in the calibration of thermometers and scientific instruments.
In applied sciences, understanding this concept is vital because it provides a specific temperature at which substances can naturally transition between phases without an external energy input or atmospheric change.

Absolute Temperature Scale

An absolute temperature scale is one based on thermodynamic principles where the zero point is absolute zero. Absolute zero represents the complete absence of thermal energy: it is the point at which atomic motion reaches its minimum. Two common examples of absolute temperature scales are the Kelvin and Rankine scales.
While Kelvin is often used in scientific settings, the Rankine scale utilizes the same incremental structure as Fahrenheit, which is particularly beneficial in certain engineering contexts such as thermodynamics within the United States.

• Rankine converts directly from Fahrenheit with the formula: \(\text{Temperature in Rankine} = \text{Temperature in Fahrenheit} + 459.67\).

Absolute temperature scales are essential as they allow scientists and engineers to predict energy transfer and behavior of materials more accurately. They provide a foundation for studying the laws of thermodynamics, making it easier to understand and calculate energy transformations and molecular motion within a given system.