9702 · 14.2Temperature scales

How thermometers measure temperature

0:00Put two thermometers into the same beaker of warm water. Both of them have been calibrated carefully. Both read exactly zero in melting ice and exactly one hundred in steam. One of them is a gas thermometer, and it reads 50.0 degrees Celsius. The other is a platinum resistance thermometer, and it reads fifty point four degrees Celsius. Neither thermometer is broken. So which one is right? And what does "fifty degrees" actually mean in the first place?

0:24In this video we'll find out how a thermometer measures anything at all, why two perfectly good thermometers can disagree, why physicists use a temperature scale that doesn't depend on any particular substance, how to convert between kelvin and degrees Celsius, and what the lowest possible temperature is. From the last video, you know that when two objects are in thermal contact, energy flows from the hotter one to the colder one until they reach the same temperature.

0:47At that point there's no net energy transfer between them, and they're in thermal equilibrium. That's the whole reason a thermometer works. When you put a thermometer into a cup of tea, the thermometer and the tea exchange energy until they're in thermal equilibrium. Now the thermometer is at the same temperature as the tea. So a thermometer never measures the tea directly. It measures its own temperature, and thermal equilibrium guarantees that its own temperature is the same as the tea's.

1:10But that leaves a question. How does the thermometer know its own temperature? The answer is that a thermometer uses some physical property that changes when the temperature changes. We call this a thermometric property. If you know how the property changes with temperature, then measuring the property tells you the temperature. A useful property changes noticeably for a small temperature change, has a single value at each temperature, and is easy to measure.

1:33The syllabus asks you to know four examples. Let's go through them one at a time.

Examples of thermometric properties

1:37First: the density of a liquid. This is the ordinary liquid-in-glass thermometer, filled with mercury or coloured alcohol. When the liquid warms up, it expands. The same mass of liquid now takes up more volume, so its density goes down. The extra volume has nowhere to go except up the thin tube, so the thread of liquid gets longer. The length of the thread is really showing you the change in the liquid's density.

2:00Second: the volume of a gas at constant pressure. Trap a fixed mass of gas behind a piston that can move freely, so the pressure stays the same. Heat the gas, and it expands. Cool it, and it contracts. So the volume of the gas tells you the temperature. Third: the resistance of a metal. As a metal wire gets hotter, its positive ions vibrate more, and the electrons flowing through it collide with them more often.

2:23So the resistance of the wire increases with temperature. Measure the resistance, and you have a thermometer. Platinum wire is very commonly used for this, which is why it's called a platinum resistance thermometer. Fourth: the e.m.f. of a thermocouple. Take two wires made of different metals, and join them together at two junctions. If the two junctions are at different temperatures, an e.m.f. is produced, and you can measure it with a voltmeter.

2:45The bigger the temperature difference between the junctions, the bigger the e.m.f., in general. In practice, you keep one junction at a known temperature, such as in melting ice, and put the other junction wherever you want to measure, even somewhere very hot like a flame.

Fixed points and calibration

2:59A thermometric property alone doesn't give you a number in degrees. You have to build a scale. The traditional way uses two fixed points: temperatures that are easy to reproduce anywhere in the world. The usual choices are the ice point, which is pure ice melting at standard atmospheric pressure, and the steam point, which is pure water boiling at standard atmospheric pressure. We call these zero degrees and one hundred degrees.

3:21Measure the property at each fixed point. Then assume the property changes steadily, in equal steps, between them. In other words, divide the gap into one hundred equal parts. That assumption gives a simple formula. The temperature equals the change in the property from its ice-point value, divided by the total change between the ice point and the steam point, multiplied by one hundred. Let's use it. A platinum resistance thermometer has a resistance of 100.0 ohms at the ice point and 138.5 ohms at the steam point.

3:50It's placed in a beaker of warm water, and its resistance is 119.4 ohms. What temperature does this thermometer give, on a scale that assumes resistance varies linearly with temperature? Step one: how far has the resistance risen above its ice-point value? 119.4 minus 100.0 is 19.4 ohms. Step two: what is the full rise from the ice point to the steam point? 138.5 minus 100.0 is 38.5 ohms. That full rise of 38.5 ohms corresponds to one hundred degrees.

4:24Step three: our rise is 19.4 out of 38.5 of the way from the ice point to the steam point. So the temperature is 19.4 divided by 38.5, times one hundred, which is about 50.4 degrees Celsius. And that's our mystery thermometer from the start. The beaker was the same one where the gas thermometer read 50.0 degrees. So now we need to understand why they disagree.

Why thermometers disagree

4:48Here's the key point. When we built that scale, we assumed the resistance of platinum rises in equal steps for equal rises in temperature. That's an assumption about the material, not a law of nature. In reality, the resistance of platinum doesn't vary exactly linearly with temperature. Its graph is very slightly curved. So when you draw a straight line between the two fixed points and read temperatures off it, you get a slightly wrong answer everywhere except at the fixed points themselves.

5:12The same is true of every thermometric property. Liquids don't expand perfectly evenly. The e.m.f. of a thermocouple is noticeably non-linear. And each property is non-linear in its own different way. So if you calibrate four thermometers, based on four different properties, at the same two fixed points, they will all agree at zero and at one hundred, because you forced them to. But in between, and outside that range, they can give different readings for the same temperature.

5:36Water gives an extreme example. Water is densest at about four degrees Celsius. Warm it from zero up to four degrees and it actually contracts. Above four degrees, it expands. So a thermometer based on the density of water would give the same reading at about two degrees and at about six degrees. A single reading could mean two different temperatures, which makes water useless as a thermometric liquid in that range.

5:59So a scale built on a particular property is really a scale of that property. Change the substance, and you change the scale. Physicists solved this by defining a scale that doesn't rely on any substance at all: the thermodynamic temperature scale, measured in kelvin. The thermodynamic scale is based on the laws of physics about energy, not on how any particular material happens to behave. It doesn't assume that mercury expands evenly or that platinum's resistance rises evenly.

6:24That's why the syllabus says it does not depend on the property of any particular substance. Thermodynamic temperature is one of the SI base quantities, and its unit, the kelvin, is an SI base unit. But real thermometers still use real substances, so how do they connect to it? It turns out that gases at low pressure behave almost identically to each other, whichever gas you use, and very close to an ideal gas.

6:47A thermometer based on an ideal gas gives temperatures that match the thermodynamic scale. So a gas thermometer is a very good practical reference, and that's why, in our beaker, the gas thermometer's 50.0 degrees was the better answer. Other thermometers can be calibrated against it, so that their readings are corrected to the thermodynamic scale. You'll use the link between gases and thermodynamic temperature a lot in the next chapter, on ideal gases.

Absolute zero and kelvin

7:11The thermodynamic scale has a natural starting point: absolute zero. Imagine cooling a gas at constant pressure and plotting its volume against its temperature in degrees Celsius. You get a straight line, with the volume decreasing steadily as you cool it. Now extend that line backwards, beyond where you've measured. It reaches zero volume at about minus 273 degrees Celsius. Try a different gas, or a different amount of gas, and the line has a different slope, but it reaches zero at the same temperature.

7:38A real gas would turn into a liquid long before it got there, so this is an extrapolation, not something you can measure directly. But it points to something real. There's a lowest possible temperature, at which the particles of a substance have the minimum possible energy. You can't remove any more energy from them, so you can't make anything colder. This lowest possible temperature is called absolute zero. On the thermodynamic scale, it's zero kelvin.

8:02It's also minus 273.15 degrees Celsius. Because nothing can be colder than absolute zero, there are no negative temperatures on the kelvin scale. Scientists have cooled things to within a tiny fraction of a kelvin of absolute zero, but never below it. In everyday life we use degrees Celsius, but in physics equations you'll almost always need kelvin. So you need to convert between them. Here's the relationship you must recall: The temperature in kelvin equals the temperature in degrees Celsius plus 273.15.

8:32Capital T is thermodynamic temperature, and the Greek letter theta is temperature in degrees Celsius. The slashes mean "divided by the unit", so T over kelvin is just the number you'd write in kelvin. The equation connects the numbers on the two scales. Why does this simple addition work? Because the two scales have exactly the same size of step. A rise of one kelvin is exactly the same as a rise of one degree Celsius.

8:56The only difference between the scales is where they put their zero. The Celsius zero sits 273.15 steps above absolute zero, so you just shift every reading by 273.15. Think of two rulers with identical markings, where one ruler has been slid along by 273.15 units. Every mark lines up with a mark on the other ruler, and the readings always differ by the same amount.

Kelvin conversions and exam practice

9:19Let's do some conversions in both directions. First, a warm room at 25 degrees Celsius. Add 273.15. That's 298.15 kelvin. Second, liquid nitrogen boils at about minus 196 degrees Celsius. Add 273.15 to minus 196. That gives 77.15 kelvin, or about 77 kelvin. Notice that a negative Celsius temperature still gives a positive number of kelvin, as it must. Now the other direction. To go from kelvin to degrees Celsius, rearrange the equation and subtract 273.15.

9:56A human body at about 310 kelvin. Subtract 273.15, and you get 36.85 degrees Celsius, which is about 37 degrees, just as you'd expect. And absolute zero, zero kelvin. Zero minus 273.15 is minus 273.15 degrees Celsius. In many calculations, you'll see 273 used instead of 273.15, and that's usually fine when the data are only given to two or three significant figures. But the exact relationship is plus 273.15, and that's the one to recall if you're asked for it.

10:32Because the steps on the two scales are the same size, a temperature difference is the same number in kelvin as in degrees Celsius. Suppose water is heated from 20 degrees Celsius to 80 degrees Celsius. The rise is 60 degrees Celsius. In kelvin, the start is 293.15 kelvin and the end is 353.15 kelvin. The rise is still 60 kelvin. The 273.15 appears in both temperatures, so it cancels when you subtract.

11:00So a change in temperature of 60 degrees Celsius is a change of 60 kelvin. You never add 273.15 to a temperature change. But ratios are a different story. Heating water from 20 degrees Celsius to 40 degrees Celsius does not double its temperature. In kelvin, that's 293.15 to 313.15, which is an increase of only about 7 percent. Doubling a temperature only has physical meaning on the thermodynamic scale, because only the kelvin scale starts from a true zero.

11:28That's why every gas equation in the next chapter uses temperature in kelvin. Let's go through the mistakes that come up most often. Mistake one: adding 273 to a temperature change. A rise of 15 degrees Celsius is a rise of 15 kelvin, not 288 kelvin. The offset only applies to an actual temperature, never to a difference between two temperatures. Mistake two: saying "degrees kelvin". The unit is simply the kelvin, with the symbol capital K and no degree sign.

11:57You say 300 kelvin, not 300 degrees kelvin. Mistake three: thinking that two correctly calibrated thermometers must always agree. They only have to agree at the fixed points used to calibrate them. Everywhere else, they can differ, because each thermometric property varies non-linearly with temperature in its own way. Mistake four: using degrees Celsius when an equation needs thermodynamic temperature. If a relationship involves temperature itself, rather than a temperature change, it needs kelvin.

12:26Using Celsius in something like a gas equation gives the wrong answer, and a Celsius temperature of zero or below would give nonsense. Here's a typical exam-style question. "A thermocouple thermometer and a resistance thermometer are both calibrated at the ice point and the steam point. Explain why they may give different readings when used to measure the same temperature of 50 degrees Celsius." Build the answer one link at a time.

12:49First link: each thermometer uses a different thermometric property, the e.m.f. of the thermocouple and the resistance of the metal. Second link: the scale assumes that each property varies linearly with temperature between the fixed points. Third link: in reality, the properties don't vary linearly with temperature, and they vary in different ways. So the two thermometers agree at the fixed points, but can disagree between them. Words like "non-linear" and "fixed points" are what earn the marks here.

13:15Here's a shorter calculation-style question. "The temperature of a gas rises from 27 degrees Celsius to 327 degrees Celsius. State the change in temperature in kelvin, and show that the thermodynamic temperature has approximately doubled." The change is 327 minus 27, which is 300 degrees Celsius, so it's 300 kelvin. No conversion needed for a difference. For the doubling, you must convert each temperature. 27 degrees Celsius is 300.15 kelvin, and 327 degrees Celsius is 600.15 kelvin.

13:47600.15 divided by 300.15 is very nearly two. If you use 273 instead of 273.15, you get exactly 300 kelvin and 600 kelvin, which makes the doubling obvious. In Celsius, 327 looks twelve times bigger than 27, which is misleading, because the Celsius scale doesn't start from absolute zero. Let's pull it all together. A thermometer reaches thermal equilibrium with an object, then uses a physical property that varies with temperature: for example the density of a liquid, the volume of a gas at constant pressure, the resistance of a metal, and the e.m.f.

14:23of a thermocouple. A scale built from a property uses fixed points, like the ice point and the steam point, and assumes the property varies linearly between them. Because real properties vary non-linearly, each in its own way, thermometers based on different properties agree at the fixed points but can disagree in between. The thermodynamic temperature scale does not depend on the property of any particular substance. Its unit is the kelvin, and gas thermometers at low pressure are a good practical match to it.

14:50The lowest possible temperature is zero kelvin, called absolute zero, which is minus 273.15 degrees Celsius. To convert, T in kelvin equals theta in degrees Celsius plus 273.15. A temperature difference is the same number in kelvin and in degrees Celsius, but ratios of temperatures only make sense in kelvin. In the next video, we'll use temperature changes to work out how much energy it takes to heat things up and to melt or boil them: specific heat capacity and specific latent heat.