Heat. Temperature. Fahrenheit and Centigrade temperature scales. Coefficient of linear expansion. Thermal stresses.

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Def. Heat. 1. A form of energy directly associated with and proportional to the random molecular motions of a substance or body as caused by combustion, friction, chemical action, radiation, etc. and convertible into other forms of energy. 2. That which raises the temperature of a body or substance. 3. The sensation produced by a rise in temperature.

Funk & Wagnalls Dictionary

Up until the mid 1800's heat was viewed as an invisible, weightless substance called caloric that passed from a hot to a cold body. It is now known to be a form of energy associated with the random molecular motion of molecules.

Sources of heat

1. The sun. Directly of indirectly nearly all of our heat can be traced to the sun. Light is necessary to the growth of plants. Coal, oil and natural gas are fossil fuels whose origin is ancient plants.

2. The earth’s interior. Geothermal energy, as from geysers.

3. Chemical action. Burning of coal, oil, gas and wood.

4. Mechanical energy. Waterpower can be converted into electricity which can be converted into heat.

5. Nuclear energy. Nuclear energy can be converted into electricity and then into heat.

Things heat can do

1. As heat is absorbed by a substance, the temperature of the substance generally rises. Water in a pan increases in temperature when it is heated.

2. The heat absorbed by a solid may cause the solid to melt, or to change from the solid to the liquid phase. Ice melts when it is heated.

3. A liquid that is heated may absorb enough heat to cause it to change from a liquid state to a vapor i.e. evaporate. Water, sufficiently heated, changes to steam.

4. The volume of almost any object increases when it is heated. The mercury in a thermometer expands as the temperature rises.

5. Heat causes many chemical reactions. During the cooking of food we apply heat to produce chemical reactions.

Temperature. Temperature is a measure of the warmness or hotness of something as measured by a thermometer. The thermometer is an invention that allows one to assign numbers to various degrees of warmness. Thus a temperature of 70^o F corresponds to that degree of warmth that we associate with that temperature. The most common thermometer is the mercury thermometer which utilizes the fact that mercury expands with increasing temperature.

Centigrade temperature scale. In the centigrade scale the freezing point of water is assigned a value of 0 and the boiling point (steam-point) of water at one atmosphere of pressure is assigned a value of 100. The space between these two points is then graduated into one hundred equal parts called degrees with the graduations numbered from 0 to 100. The graduations are then continued below 0 and above 100. The ice-point (temperature of melting ice) and steam-point of water represent two fixed points that are easily reproduced.

Fahrenheit temperature scale. In the Fahrenheit scale the ice-point of water is assigned a value of 32 and the steam-point at one atmosphere of pressure is assigned a value of 212. The space between is then graduated into 180 equal parts called degrees with the graduations going both below 32 and above 180.

The fixed points of the Fahrenheit scale were originally based on the temperatures of a mixture of ice and salt and the temperature of the human body. In this system the ice-point of water happened to be 32^o and the temperature of boiling water was approximately 212^o.

Conversion between Fahrenheit and Centigrade temperatures. Since 100 Centigrade degrees equal 180 Fahrenheit degrees, one Centigrade degree equals 180/100 = 9/5 Fahrenheit degrees.

1. To change Centigrade readings to Fahrenheit readings, multiply the Centigrade readings by 9/5 and add 32.

2. To change Fahrenheit readings to Centigrade readings, subtract 32 from the Fahrenheit readings and multiply by 5/9.

In formula form, these procedures are

where t_F is the Fahrenheit temperature and t_C is the Centigrade temperature.

Thermometers. The usual mercury thermometer is generally limited by the freezing point and boiling point of mercury, 39^o C and 357^o C, respectively. Other types of thermometers have been devised for temperatures outside this range. An alcohol thermometer is useful for temperatures below the range of the mercury thermometer. Constant volume gas thermometers are useful over a wide range of temperatures. The platinum resistance thermometer makes use of the fact that the electrical resistance of metals increases when their temperature is increased and is one of the most precise thermometers.

Our sense of temperature. Consider the following experiment: Take three glasses and fill one with hot water, one with warm water and one with ice water. Put them side by side on a table with the one with hot water on the left, the one with warm water in the middle, and the one with the ice water on the right. For a few seconds hold the index finger of your left hand in the hot water and the index finger of your right hand in the ice water. Then put both fingers in the glass of warm water. The warm water will feel cold to your left hand and warm to your right hand. Thus our sense of temperature can be inaccurate and misleading. If you walk into a bathroom with a tile floor on a cold morning the floor will feel cold. If part of the floor is covered by a rug and you step on the rug, the rug will seem much warmer. Yet a thermometer will show that the floor and rug are the same temperature. Why does the tile feel a lot colder than the rug? Because the tile conducts heat away from your feet much more rapidly than does the rug. From this we see human touch is easily fooled in regard to temperature.

Linear expansion of solids and liquids. Most substances, with very few exceptions, expand when they are heated, with different substances expanding at different rates. The increase in length of a solid object due to an increase in temperature is proportional to the original length and to the increase in temperature i.e.

Δl = αlΔt

where

l = original length

Δl = increase in length

Δt = change in temperature

α = proportionality constant, different for different materials, called the coefficient of linear expansion

The above result, although not always totally true, is valid over a fairly large range of temperatures.

Def. Coefficient of linear expansion. The fractional increase in length per degree rise in temperature.

Area expansion. The increase in area due to an increase in temperature is given by

ΔA = γAΔt

where

A = the original area

ΔA = increase in area

Δt = change in temperature

γ = coefficient of area expansion

Volume expansion. The increase in volume due to an increase in temperature is given by

ΔV = βVΔt

where

V = the original volume

ΔV = increase in volume

Δt = change in temperature

β = coefficient of volume expansion

Because solids expand in all directions when they are heated, the coefficient of area expansion γ, or the increase in unit area per degree, is approximately twice the coefficient of linear expansion. Similarly, the coefficient of volume expansion β, or the increase in unit volume per degree, is approximately three times the coefficient of linear expansion.

Thermal stresses. If the ends of a rod are rigidly fixed so as to prevent expansion or contraction and the temperature of the rod changes, stresses will be set up in the rod. These stresses, called thermal stresses, can become very large, sufficiently large to stress the rod beyond its breaking point. Thus in designing structures that are subject to changes in temperature, provision must be made for thermal expansion.

Suppose a rod at temperature t has its ends rigidly fastened and the temperature drops to a lower value t₀. The formula for the stress produced in the rod is

where

F = total force on rod

A = cross-sectional area of rod

Y = Young’s modulus

α = coefficient of linear expansion

Δt = change in temperature

Derivation. The fractional change in length of the rod if it were free to contract would be

where l is the length of the rod. Since the rod is not free to contract, the tension in the rod must increase sufficiently to produce the same fractional change in length. By Hook’s law

Thus

References

Dull, Metcalfe, Brooks. Modern Physics.

Sears, Zemansky. University Physics.