Many clinics in India still read fever in degrees Fahrenheit. A child at \(102\) has a fever, yet the weather report on the same day says \(35\) for a hot afternoon. Both numbers are temperatures. They sit on scales with different fixed-point numbers.
Three scales, one set of fixed points
Every common scale uses the same two fixed points and gives them different numbers. The Fahrenheit scale calls the ice point \(32\,°\text{F}\) and the steam point \(212\,°\text{F}\), so it has 180 degrees between them. The Kelvin scale calls the ice point \(273.15\,\text{K}\) and the steam point \(373.15\,\text{K}\), 100 kelvin apart. For most sums the ice point is taken as \(273\,\text{K}\), as in this chapter unless precision matters.
All three scales are straight lines between the same two points. So the fraction of the way from the ice point to the steam point is the same number on each scale. Writing that fraction three times gives the conversion rule.
\[ \dfrac{C - 0}{100} = \dfrac{F - 32}{180} = \dfrac{K - 273}{100} \]
Move the slider to \(37\,°\text{C}\), the normal temperature of the body. The Fahrenheit reading is \(\tfrac95 \times 37 + 32 = 98.6\,°\text{F}\). The fever of \(102\,°\text{F}\) is \((102 - 32) \times \tfrac59 = 38.9\,°\text{C}\). One column, two numbers: the scales differ in their labels, never in the temperature.
On a May afternoon in Phalodi, Rajasthan, the thermometer reads \(48\,°\text{C}\). Write this temperature in °F and in K. Find also the temperature at which the Celsius and Fahrenheit readings are the same number.
(a) \(F = \tfrac95 C + 32 = \tfrac95 \times 48 + 32 = 86.4 + 32 = 118.4\,°\text{F}\).
(b) \(K = C + 273 = 48 + 273 = 321\,\text{K}\).
(c) Put \(F = C\) in \(F = \tfrac95 C + 32\). Then \(-\tfrac45 C = 32\), so \(C = -40\). At \(-40\,°\text{C}\) the Fahrenheit thermometer also reads \(-40\,°\text{F}\).
A clinical thermometer reads \(104\,°\text{F}\). What is this temperature in °C?
At what Celsius temperature is the Fahrenheit reading exactly twice the Celsius reading?
A zero that is not arbitrary
The zero of the Celsius scale is only the melting point of ice. Nothing special happens to the rest of nature there. The Fahrenheit zero is even more arbitrary. Is there a temperature that is a true zero, one set by nature?
Take a fixed amount of any gas in a closed rigid bulb, so its volume cannot change, and measure its pressure at several temperatures. The pressure rises along a straight line. It rises by \(1/273.15\) of its ice-point value for each degree. Three gases in Fig 2.1 start at three different pressures.
Extend the three lines to lower temperatures. All three reach zero pressure at the same temperature, \(-273.15\,°\text{C}\). A real gas turns to liquid before it gets there, but the point is the same for every gas. This temperature is absolute zero. No body can be cooled below it.
The Kelvin scale
The Kelvin scale puts its zero at absolute zero and keeps the Celsius degree. So \(T = t + 273.15\), where \(T\) is in kelvin and \(t\) in °C. A kelvin is exactly as large as a Celsius degree. So a rise of \(25\,°\text{C}\) is a rise of \(25\,\text{K}\), and it is a rise of \(25 \times 1.8 = 45\) Fahrenheit degrees. No temperature on the Kelvin scale is negative, because nothing is colder than its zero.
Absolute zero is \(-273.15\,°\text{C}\), which is \(\tfrac95 \times (-273.15) + 32 = -459.67\,°\text{F}\). Liquid nitrogen boils at \(-196\,°\text{C}\), or \(77\,\text{K}\).
A ratio of temperatures means something only in kelvin. A day at \(20\,°\text{C}\) is not twice as hot as a day at \(10\,°\text{C}\). In kelvin the two are \(293\) and \(283\), a ratio of about \(1.04\). A temperature difference, on the other hand, is the same number in °C and in K.
The air in a room warms from \(15\,°\text{C}\) to \(35\,°\text{C}\). What is the rise in temperature in kelvin and in Fahrenheit degrees?
One body is at \(27\,°\text{C}\) and another at \(327\,°\text{C}\). A student says the second is about twelve times as hot as the first. Is the ratio meaningful? What is the ratio of their absolute temperatures?
The Celsius ratio \(327/27\) means nothing, because the Celsius zero is an arbitrary point. Only temperatures measured from absolute zero can be compared as ratios. In kelvin the two are \(300\,\text{K}\) and \(600\,\text{K}\), so the second is at twice the absolute temperature of the first.
Any scale at all
Anyone can make a new scale by choosing two numbers for the fixed points. The old Réaumur scale, once used in Europe, called them \(0\) and \(80\). Suppose a scale \(X\) marks the ice point \(X_\text{ice}\) and the steam point \(X_\text{steam}\). The fraction rule of the three common scales holds for it too.
A scale \(Y\) marks the ice point \(10\,°\text{Y}\) and the steam point \(160\,°\text{Y}\). Find the reading on this scale at \(40\,°\text{C}\), and the Celsius temperature when it reads \(100\,°\text{Y}\). How many Celsius degrees make one degree \(Y\)? Convert also \(40\) degrees Réaumur to °C and °F.
(a) \(\dfrac{Y - 10}{150} = \dfrac{40}{100}\), so \(Y = 10 + 150 \times 0.4 = 70\,°\text{Y}\).
(b) \(C = 100 \times \dfrac{100 - 10}{150} = 60\,°\text{C}\).
(c) \(150\) degrees \(Y\) fill the same gap as \(100\) Celsius degrees, so one degree \(Y\) is \(\tfrac{100}{150} = 0.67\) Celsius degree.
(d) Réaumur has \(80\) degrees between the fixed points, so \(C = 40 \times \tfrac{100}{80} = 50\,°\text{C}\), and \(F = \tfrac95 \times 50 + 32 = 122\,°\text{F}\).
A scale marks the ice point as \(-20\) and the steam point as \(180\). It reads \(30\). What is the Celsius temperature?
A thermometer with a wrong scale
A thermometer whose scale was painted in the wrong place is called faulty. Say it reads \(-1\) in melting ice and \(103\) in steam. Its column still rises steadily, so it is simply a scale with fixed points \(-1\) and \(103\). The fraction rule converts its readings like those of any other scale.
For this thermometer the reading \(R\) at a true temperature \(t\) is \(R = -1 + 1.04\,t\), because \(104\) of its degrees span \(100\) Celsius degrees. A reading of \(60\) means \(t = (60 + 1)/1.04 = 58.7\,°\text{C}\). In the sim above it read \(61.4\) at a true \(60\,°\text{C}\).
The error in Fig 2.2 is a straight line from \(-1\) to \(+3\). It must cross zero once. Set the reading equal to the true temperature: \(t = -1 + 1.04\,t\), so \(0.04\,t = 1\) and \(t = 25\,°\text{C}\). At \(25\,°\text{C}\), and only there, this thermometer tells the truth.
A thermometer reads \(-1\) in melting ice and \(103\) in steam. What does it read when the true temperature is \(40\,°\text{C}\)? At what temperature does it read correctly?
The faulty scale has \(103 - (-1) = 104\) divisions between the fixed points, for \(100\) Celsius degrees.
\(\dfrac{R - (-1)}{104} = \dfrac{t}{100}\), so \(R = -1 + 1.04\,t\).
At \(t = 40\,°\text{C}\), \(R = -1 + 41.6 = 40.6\). For a correct reading put \(R = t\): \(0.04\,t = 1\), \(t = 25\,°\text{C}\).
It reads \(40.6\) at a true \(40\,°\text{C}\), and it is correct only at \(25\,°\text{C}\).
A faulty thermometer reads \(2\) in melting ice and \(98\) in steam. It reads \(50\) in a bath. What is the true temperature of the bath, in °C?
A faulty thermometer reads \(5\) in melting ice and \(95\) in steam. Which statement is correct?
In 1848 the young Glasgow professor William Thomson proposed an absolute scale of temperature, one that did not depend on any one substance. He based it on the working of an ideal heat engine. In the same paper he noted that the air thermometer points to a lowest possible temperature. It lies near \(-273\,°\text{C}\), where the gas lines of this lesson meet. Thomson later became Lord Kelvin, and the scale that counts from that zero carries his name.
A skin doctor removes a wart by touching it with liquid nitrogen for a few seconds. The liquid boils at \(-196\,°\text{C}\), which is only \(77\) kelvin above absolute zero. Blood banks and IVF laboratories in Indian cities store cells in tanks of it. At \(77\,\text{K}\) the chemistry of a cell almost stops, and the cell can be revived years later.
- Converting a temperature difference by adding \(273\) or \(32\).A difference of \(1\,°\text{C}\) is \(1\,\text{K}\) and \(1.8\) Fahrenheit degrees. The offsets cancel in a difference.
- Forgetting to subtract \(32\) before scaling a Fahrenheit reading.\(C = \tfrac59 (F - 32)\). Scale the distance from the ice point, never the reading itself.
- Taking a ratio of Celsius temperatures.Ratios need absolute temperatures. Convert to kelvin first.
- Thinking a faulty thermometer is wrong everywhere.Its error runs along a straight line and crosses zero at one temperature.