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 \(\left(0^{\circ} \mathrm{R}\right)\) 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

491.688 °R

Step 1: Understanding the Triple-Point of Water

The triple-point of water is the temperature and pressure at which water can exist in equilibrium in all three states: solid, liquid, and gas. Its temperature is precisely defined as 0.01°C or 273.16K on the Kelvin scale.

Step 2: Converting Kelvin to Rankine

We know that 1 K is the same temperature change as 1 °R because both Kelvin and Rankine scales are absolute, but Rankine uses the same increment as Fahrenheit rather than Celsius. The relation between Kelvin (K) and Rankine (°R) is given by: \[ 1 K = 1.8 °R \]Thus, to convert from Kelvin to Rankine, we use the formula: \[ T(°R) = T(K) \times 1.8 \]

Step 3: Calculating the Triple-Point Temperature in Rankine

Substitute the triple-point temperature of water in Kelvin into the conversion formula: \[ T(°R) = 273.16 \times 1.8 \] Carrying out the multiplication: \[ T(°R) = 491.688 \]So, the triple-point temperature of water on the Rankine scale is 491.688 °R.

Key Concepts

Absolute Temperature Scale

An absolute temperature scale is one in which the zero point is the lowest possible temperature, known as absolute zero. Absolute zero is the point at which all molecular motion stops. On such a scale, temperatures are measured in units that directly relate to the fundamental properties of thermodynamics.

In practical terms, absolute temperature scales are used to describe temperature ranges that start from absolute zero. The Kelvin and Rankine scales are the most commonly used absolute temperature scales.

• The Kelvin scale uses Celsius-sized units (°C), but starts at absolute zero, which is 0 K.
• The Rankine scale uses Fahrenheit-sized units (°F), also starting at absolute zero, labeled as 0 °R.

Both scales share the characteristic that they measure temperature in relation to thermodynamic laws, offering an essential basis for scientific calculations and formulations.

Triple-Point of Water

The triple-point of water is a crucial concept in thermodynamics. It's the specific set of conditions where water exists in equilibrium as solid ice, liquid water, and gas (vapor). This delicate balance occurs at a precise temperature and pressure.

On the Kelvin scale, the triple-point temperature is 273.16 K, while on the Celsius scale, it's marked as 0.01°C. At the triple-point, water can coexist in all three states, making it an important reference point for temperature scales.

• It serves as a standard calibration point for thermometers.
• Helps establish the definitions for many temperature scales, including the Kelvin and Rankine.

Understanding this concept is important for deep dives into thermodynamics and ensures accurate measurements across scientific fields.

Kelvin to Rankine Conversion

Converting measurements between the Kelvin and Rankine scales involves a few key steps, owing to the differences in unit size these scales use. While both scales start at absolute zero, they differ in increments: Kelvin uses Celsius increments, and Rankine uses Fahrenheit increments.

The formula to convert Kelvin (K) to Rankine (°R) is given by:
\[ T(°R) = T(K) \times 1.8 \]
Here, the factor of 1.8 accounts for the larger size of Fahrenheit degrees compared to Celsius degrees.

• The same amount of temperature in Kelvin is 1.8 times when expressed in Rankine.
• This conversion is straightforward multiplication since both scales are absolute.

This conversion allows for the seamless transition of temperature values in scientific settings where different measuring standards are applied.

Thermodynamic Temperature Scales

Thermodynamic temperature scales are designed based on principles of thermodynamics, which provide a broad understanding of heat, work, and energy. These scales measure temperature in a way that's consistent with the laws of thermodynamics, ensuring reliable and universal results in scientific processes.

Kelvin and Rankine are both thermodynamic temperature scales, used extensively in physics and engineering.

• The Kelvin scale is part of the International System of Units (SI) and is preferred in most scientific work worldwide.
• The Rankine scale is often used in engineering in the United States, particularly when dealing with thermodynamic cycles such as those in engines.

These scales allow scientists and engineers to make connections between various phenomena and create models for energy systems efficiently. By using absolute measurements, they facilitate clear and consistent communication of thermal data.