Heat and Temperature

Put a hot glass of tea on a cold marble table and, after a while, both are lukewarm. Something flowed from one to the other, and it stopped flowing when some property of the two became equal. That "something" is heat; that property is temperature. This short unit separates two words that students use as if they meant the same thing, and then shows that the rule allowing temperature to be measured at all is a law of physics in its own right — the zeroth law.

9.1 Molecular concept of thermal energy, heat and temperature; cause and direction of heat flow

Every substance is made of molecules that are never at rest. In a solid they vibrate about fixed lattice positions; in a liquid they vibrate and slide past one another; in a gas they fly freely and collide. Each molecule therefore has kinetic energy, and because the molecules attract one another they also have potential energy.

Temperature is not the total energy but the average energy per molecule. For an ideal gas the kinetic theory gives the average translational kinetic energy of one molecule as

where k_{B} = 1.38\times10^{-23}\ \text{J K}^{-1} is the Boltzmann constant and T is the absolute temperature in kelvin. Turn this around and it says something important: temperature is a measure of the average kinetic energy of the molecules of a body.

Cause and direction of heat flow. When two bodies at different temperatures touch, their surface molecules collide. On average the faster molecules of the hotter body lose kinetic energy in these collisions and the slower molecules of the colder body gain it. The net result is a transfer of energy from the body at higher temperature to the body at lower temperature. So:

• The cause of heat flow is the temperature difference, not a difference in the amount of heat, mass or size.
• The direction of heat flow is always from higher to lower temperature.
• Flow stops when the temperatures become equal — not when the energies become equal.

9.2 Thermal equilibrium and Zeroth law of thermodynamics

Two bodies placed in thermal contact exchange energy until the net flow becomes zero. They are then said to be in thermal equilibrium, and experiment shows that this happens exactly when their temperatures are equal.

Now suppose body A is in equilibrium with body C, and body B is separately in equilibrium with the same body C. Is A then in equilibrium with B? Experiment says yes, always. This result cannot be deduced from the first or second laws, and it is so basic that it was named the zeroth law after the others had already been numbered.

The zeroth law is the licence to use a thermometer. A thermometer is the third body C. We bring it into contact with A and read it, then with B and read it. If the readings agree, the law guarantees that A and B are in equilibrium with each other — so the reading is a genuine property of the state of each body. That property is what we call temperature.

9.3 Thermal equilibrium as working principle of mercury thermometer

Any physical property that changes measurably and reproducibly with temperature can be used to measure temperature. Such a property is called a thermometric property — the length of a liquid column, the pressure of a fixed mass of gas, the resistance of a wire, the e.m.f. of a thermocouple.

A mercury thermometer is a thin-walled glass bulb joined to a uniform capillary tube of very fine bore, sealed after the air above the mercury has been removed. The bulb holds the mercury; the fine bore makes a small expansion of the mercury produce a long, easily read movement of the thread.

How it works — thermal equilibrium is the whole principle. The bulb is placed in contact with the body whose temperature is wanted. Heat flows between body and bulb until the two reach thermal equilibrium, i.e. until the mercury is at the same temperature as the body. The mercury, having a larger expansivity than glass, has by then moved along the stem to a definite position, and by the zeroth law that position is a measure of the body's own temperature. Note the two conditions hidden in this: the thermometer must be left long enough to reach equilibrium, and it must be small enough that the heat it absorbs does not noticeably change the body's temperature.

Calibration. Two fixed points are used:

• Lower fixed point (ice point): the temperature of pure melting ice at standard atmospheric pressure, marked 0\ ^{\circ}\text{C}; the thread length is l_0.
• Upper fixed point (steam point): the temperature of steam above water boiling at standard atmospheric pressure, marked 100\ ^{\circ}\text{C}; the thread length is l_{100}.

The interval between them is divided into 100 equal parts, which gives the working formula of every two-fixed-point thermometer.

The three scales in use are related in exactly the same way — each fraction is "how far along its own scale", and equal fractions mean the same temperature:

A clinical thermometer is the same instrument with the range narrowed to about 35\ ^{\circ}C–42\ ^{\circ}C so that each division can be 0.1\ ^{\circ}C, and with a constriction in the bore just above the bulb. The constriction lets mercury push past while the thread rises but breaks the thread when it cools, so the reading is held until the thermometer is shaken down.

Practice questions

Group A — Multiple choice (1 mark each)

1. Heat flows from body A to body B when placed in contact because A has [1]amore heatbgreater masschigher average molecular kinetic energydgreater volume
2. The law of thermodynamics that makes temperature a measurable quantity is the [1]azeroth lawbfirst lawcsecond lawdthird law
3. The temperature at which the Celsius and Fahrenheit scales read the same is [1]a0^{\circ}b-32^{\circ}c-40^{\circ}d100^{\circ}
4. The thermometric property of a mercury-in-glass thermometer is the [1]apressure of mercurybresistance of mercuryclength of the mercury columndmass of mercury
5. A temperature of 300\ \text{K} is approximately [1]a27\ ^{\circ}Cb300\ ^{\circ}Cc573\ ^{\circ}Cd-27\ ^{\circ}C

Group B — Short answer (5 marks each)

1. Distinguish between heat and temperature on a molecular basis. Explain why a bucket of warm water can hold more thermal energy than a small flame at a much higher temperature. [5]
2. State the zeroth law of thermodynamics and explain how it makes the measurement of temperature possible. [5]
3. Explain thermal equilibrium as the working principle of a mercury thermometer, and give four reasons why mercury is preferred to water as a thermometric liquid. [5]
4. The mercury thread in a thermometer is 4.0\ \text{cm} long at the ice point and 24.0\ \text{cm} long at the steam point. Find(a) the temperature when the thread is 13.0\ \text{cm} long and(b) the thread length at 65\ ^{\circ}\text{C}. [5]
5. Calculate the average translational kinetic energy of a gas molecule at 127\ ^{\circ}\text{C}, and find the temperature in ^{\circ}C at which it is half this value. [5]

Group C — Long answer (8 marks each)

1. (a) Explain, on the molecular model, what is meant by thermal energy, heat and temperature, and state the cause and direction of heat flow. [4](b) State the zeroth law of thermodynamics and describe the construction and working of a mercury-in-glass thermometer, showing how it is calibrated between the two fixed points. [4]
2. (a) Define thermometric property and name the property used in a constant volume gas thermometer and in a platinum resistance thermometer. [2](b) A faulty thermometer reads 2\ ^{\circ}\text{C} in melting ice and 102\ ^{\circ}\text{C} in steam. What is the true temperature when it reads 77\ ^{\circ}\text{C}? [3](c) At what temperature does the Fahrenheit reading become exactly twice the Celsius reading? [3]