Temperature and Thermometry

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle to start. Take two thermometers, one in degrees Celsius and one in degrees Fahrenheit, and put both into the same glass of water. Is there any temperature at which both show exactly the same number? Hold that thought; we will settle it today. Today: temperature and thermometry. Conversions between scales come up often in JEE, so let's begin.

0:27What temperature tells us

So, let us start simple. Put a cold steel spoon into a cup of hot tea. Which way does heat flow: from the tea into the spoon, or the other way? From the tea into the spoon. Yes, exactly! Heat flows from the hotter body to the colder one, and temperature tells us which is hotter. So temperature decides the direction of heat flow. Okay? Now wait a while. The spoon warms up, the tea cools a little, and the net flow of heat stops. The two are now in thermal equilibrium, and bodies in thermal equilibrium have equal temperatures. The zeroth law of thermodynamics is what makes this work: if A and B are each in thermal equilibrium with a third body C, then A and B are in thermal equilibrium with each other. That third body is just a thermometer! Right?

1:20Three scales, two fixed points

Now, every temperature scale is built on two fixed points. The first is the ice point, the temperature of melting ice at normal atmospheric pressure. Zero degrees Celsius: the small circle means degrees, and capital C is Celsius. Thirty two degrees on the Fahrenheit scale, capital F, and two seventy three point one five on the kelvin scale. Capital K has no little circle: we say kelvin, not degrees kelvin. Got it? The second is the steam point, steam over boiling water: one hundred degrees Celsius, two hundred and twelve Fahrenheit, and three seventy three point one five kelvin. With me? Now look at the gap between these two fixed points, on each scale. How many divisions does each scale put in that gap? Let's see. Celsius: one hundred. Fahrenheit: two hundred and twelve minus thirty two, that is one hundred and eighty! And kelvin: one hundred, just like Celsius. Clear? So here is the conversion formula. C over one hundred, equals F minus thirty two over one hundred and eighty, equals K minus two seventy three point one five over one hundred. Each fraction says how far you have climbed above the ice point, as a fraction of the whole gap. Okay? From this, F equals nine fifths C, plus thirty two; nine fifths is just one eighty over one hundred. And K equals C plus two seventy three point one five. Oh no, careful! Many students forget the plus thirty two, and then every answer is off by thirty two. Right?

3:03Any linear scale: one fraction

Now, what if a question invents a brand new linear scale of its own? Its lower fixed point is capital L, and its upper fixed point is capital U, okay? So how do we connect this new scale to Celsius? Think about what each fraction in our formula was measuring. Yes, exactly! How far you are above the lower fixed point, as a fraction of the whole gap. Call the reading on the new scale capital X, and then X minus L, over U minus L, is the same on every scale. One condition: the scale must be linear, equal steps for equal temperature changes. With me? Quick check with Celsius: L is zero and U is one hundred. The double arrow means gives: C minus zero over one hundred minus zero, which is just C over one hundred. Perfect! Keep this fraction ready for Example two. Got it?

3:59The SI scale: triple point of water

Now, the S I scale, the kelvin scale. It is fixed by one single point: the triple point of water, the one state in which ice, liquid water and water vapour all exist together. Right? Quick question. The ice point is two seventy three point one five kelvin. Is the triple point the same number? No, no, careful! The triple point of water is two seventy three point one six kelvin. Point one six, okay? The ice point, two seventy three point one five, is one hundredth of a kelvin lower. Mixing up these two points is a common trap. Point one six fixes the S I scale; point one five goes in our conversion formula. Clear?

4:44Example 1: where Celsius equals Fahrenheit, and K equals F

Now, remember our puzzle? Let's settle it. At what temperature do the Celsius and Fahrenheit scales read the same? And when do Fahrenheit and kelvin agree? Pause the video, and try it first. Done? Okay, let's check. If both read the same, C equals F, so put C in place of F: C equals nine fifths C, plus thirty two. With me? Take nine fifths C to the left: minus four fifths C equals thirty two. So C equals thirty two times minus five over four: minus forty! Minus forty degrees Celsius is exactly minus forty degrees Fahrenheit. Puzzle solved, and it is a very cold one. Right? Now the second part. Call the common reading F, so the kelvin reading is F as well. Our formula links F to C, so what is C in terms of F? Yes, exactly! C equals K minus two seventy three point one five, and K is F, so C equals F minus two seventy three point one five. Okay? Now put this into F equals nine fifths C, plus thirty two. Nine fifths times two seventy three point one five is four ninety one point six seven. Minus four ninety one point six seven, plus thirty two, is minus four fifty nine point six seven. Got it? Bring nine fifths F to the left: minus four fifths F equals minus four fifty nine point six seven, so four fifths F equals four fifty nine point six seven. Multiply by five over four: F is about five seventy four point six. So at about five seventy four point six, Fahrenheit and kelvin agree. Very hot this time! One formula did both parts.

6:40Example 2: true reading of a faulty thermometer

Now a slightly harder one. Here, a faulty thermometer reads minus one degree in melting ice, and one hundred and one degrees in steam. What is the true temperature when it reads fifty degrees? Pause and try. Hint: use X minus L, over U minus L. Alright, here is a tempting shortcut. In melting ice it reads minus one instead of zero, so it reads one degree low, right? So just add one: fifty one degrees Celsius? No, no, wait! In steam it reads one hundred and one instead of one hundred: one degree high! So the error is not the same everywhere. Use the fraction. Here L is minus one, U is one hundred and one, so the gap U minus L is one hundred and two. Okay? The fraction is the same on both scales. Celsius: C minus zero, over one hundred minus zero. Faulty scale: fifty minus, minus one in brackets, over one hundred and two, which is fifty one over one hundred and two. With me? So C equals fifty one over one hundred and two, times one hundred. Fifty one over one hundred and two is exactly one half. Half of one hundred: fifty degrees Celsius! See that? At this reading, the faulty thermometer happens to be correct. One degree low at the bottom, one degree high at the top, and exact in the middle. Adding one would have cost you marks, so always use the fraction. Clear?

8:17Quick revision: conversion formula, linear scales, 273.16 K

So, quick revision. One: C over one hundred, equals F minus thirty two over one eighty, equals K minus two seventy three point one five over one hundred. Two: for any linear scale, X minus L over U minus L is the same on every scale. Three: the triple point of water, two seventy three point one six kelvin, fixes the S I scale; the ice point is two seventy three point one five. Now go and try the practice problems on this page. All the best!