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Chapter 17: Problem 10

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?

Short Answer

Expert verified
The triple-point temperature of water on the Rankine scale is 491.688掳R.

Step by step solution

01

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.
02

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.
03

Plug in the Triple-Point Temperature of Water

Now we substitute 273.16 K into the conversion formula:\[ T_{R} = 273.16 \times \frac{9}{5} \]
04

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掳R.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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 掳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掳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.

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Most popular questions from this chapter

One end of an insulated metal rod is maintained at 100.0\(^\circ\)C, and the other end is maintained at 0.00\(^\circ\)C by an ice-water mixture. The rod is 60.0 cm long and has a cross-sectional area of 1.25 cm\(^2\). The heat conducted by the rod melts 8.50 g of ice in 10.0 min. Find the thermal conductivity \(k\) of the metal.

During your mechanical engineering internship, you are given two uniform metal bars \(A\) and \(B\), which are made from different metals, to determine their thermal conductivities. Measuring the bars, you determine that both have length 40.0 cm and uniform cross-sectional area 2.50 cm\(^2\). You place one end of bar \(A\) in thermal contact with a very large vat of boiling water at 100.0\(^\circ\)C and the other end in thermal contact with an ice-water mixture at 0.0\(^\circ\)C. To prevent heat loss along the bar's sides, you wrap insulation around the bar. You weigh the amount of ice initially and find it to be 300 g. After 45.0 min has elapsed, you weigh the ice again and find that 191 g of ice remains. The ice-water mixture is in an insulated container, so the only heat entering or leaving it is the heat conducted by the metal bar. You are confident that your data will allow you to calculate the thermal conductivity \(k_A\) of bar \(A\). But this measurement was tedious-you don't want to repeat it for bar \(B\). Instead, you glue the bars together end to end, with adhesive that has very large thermal conductivity, to make a composite bar 80.0 m long. You place the free end of A in thermal contact with the boiling water and the free end of \(B\) in thermal contact with the ice-water mixture. As in the first measurement, the composite bar is thermally insulated. You go to lunch; when you return, you notice that ice remains in the ice-water mixture. Measuring the temperature at the junction of the two bars, you find that it is 62.4\(^\circ\)C. After 10 minutes you repeat that measurement and get the same temperature, with ice remaining in the ice-water mixture. From your data, calculate the thermal conductivities of bar \(A\) and of bar \(B\).

In very cold weather a significant mechanism for heat loss by the human body is energy expended in warming the air taken into the lungs with each breath. (a) On a cold winter day when the temperature is -20\(^\circ\)C, what amount of heat is needed to warm to body temperature (37\(^\circ\)C) the 0.50 L of air exchanged with each breath? Assume that the specific heat of air is 1020 J / kg \(\cdot\) K and that 1.0 L of air has mass \(1.3 \times 10{^-}{^3} kg\). (b) How much heat is lost per hour if the respiration rate is 20 breaths per minute?

Consider a poor lost soul walking at 5 km/h on a hot day in the desert, wearing only a bathing suit. This person's skin temperature tends to rise due to four mechanisms: (i) energy is generated by metabolic reactions in the body at a rate of 280 W, and almost all of this energy is converted to heat that flows to the skin; (ii) heat is delivered to the skin by convection from the outside air at a rate equal to \(k'A{_s}{_k}{_i}{_n}(T{_a}{_i}{_r} - T{_s}{_k}{_i}{_n})\), where \(k'\) is 54 J/h \(\cdot\) C\(^\circ\) \(\cdot\) m\(^2\), the exposed skin area \(A{_s}{_k}{_i}{_n}\) is 1.5 m\(^2\), the air temperature \(T{_a}{_i}{_r} \)is 47\(^\circ\)C, and the skin temperature \(T{_s}{_k}{_i}{_n}\) is 36\(^\circ\)C; (iii) the skin absorbs radiant energy from the sun at a rate of 1400 W/m\(^2\); (iv) the skin absorbs radiant energy from the environment, which has temperature 47\(^\circ\)C. (a) Calculate the net rate (in watts) at which the person's skin is heated by all four of these mechanisms. Assume that the emissivity of the skin is \(e\) = 1 and that the skin temperature is initially 36\(^\circ\)C. Which mechanism is the most important? (b) At what rate (in L/h) must perspiration evaporate from this person's skin to maintain a constant skin temperature? (The heat of vaporization of water at 36\(^\circ\)C is \(2.42 \times 10{^6}\) J/kg.) (c) Suppose instead the person is protected by light-colored clothing \((e \approx 0)\) so that the exposed skin area is only 0.45 m\(^2\). What rate of perspiration is required now? Discuss the usefulness of the traditional clothing worn by desert peoples.

On a cool (4.0\(^\circ\)C) Saturday morning, a pilot fills the fuel tanks of her Pitts S-2C (a two-seat aerobatic airplane) to their full capacity of 106.0 L. Before flying on Sunday morning, when the temperature is again 4.0\(^\circ\)C, she checks the fuel level and finds only 103.4 L of gasoline in the aluminum tanks. She realizes that it was hot on Saturday afternoon and that thermal expansion of the gasoline caused the missing fuel to empty out of the tank's vent. (a) What was the maximum temperature (in \(^\circ\)C) of the fuel and the tank on Saturday afternoon? The coefficient of volume expansion of gasoline is \(9.5 \times 10{^-}{^4} K{^-}{^1}\). (b) To have the maximum amount of fuel available for flight, when should the pilot have filled the fuel tanks?
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