TEMPERATURE. 8. Temperature and Heat 1

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1 TEMPERATURE Heat is the energy that is transferred between objects because of a temperature difference Terms such as transfer of heat or heat flow from object A to object B simply means that the total energy of A decreases and that of B increases An object does not contain heat it has a certain energy content, and the energy it exchanges with other objects due to temperature differences is called heat Objects are said to be in thermal contact if heat can flow between them When a hot object is brought into contact with a cold object, heat is exchanged The hot object cools off (its molecules move more slowly) and the cold object warms up (its molecules move more rapidly) After some time in thermal contact, the transfer of heat ceases objects are then in thermal equilibrium Thermodynamics is the study of physical processes involving the transfer of heat it deals with the flow of energy within and between objects 8. Temperature and Heat 1

2 ZEROTH LAW OF DYNAMICS Thermal equilibrium is determined by a single physical quantity temperature Two objects in thermal contact are in equilibrium when they have the same temperature If one or the other as a higher temperature, heat flows from that object to the other until their temperatures are equal The zeroth law of thermodynamics is: If object A is in thermal equilibrium with object B, and object C is also in thermal equilibrium with object B, then objects A and C will be in thermal equilibrium if brought into thermal contact 8. Temperature and Heat 2

3 THERMAL EXPANSION: LINEAR EXPANSION Most substances expand when heated In thermometers, the expansion of liquid, such as mercury or alcohol, results in a column of liquid rising (or falling) within the glass tube Water is the exception to the rule Consider a rod with length L 0 at temperature T 0 When heated or cooled, its length changes by L in direct proportion to the temperature change, T Thus L = (constant) T The constant or proportionality depends, among other things, on the substance from which the rod is made In fact experiments show that the change in length, L, is proportional to both the initial length, L 0, and the temperature change, T The constant of proportionality is referred to as α which is the coefficient of linear expansion (K -1 ) L = αl 0 T 273 Kelvin = 0 C 8. Temperature and Heat 3

4 THERMAL EXPANSION: LINEAR EXPANSION - EXAMPLE The Eiffel Tower is made from iron. If the tower is 301m high on a 22 C day, how much does its height decrease when the temperature cools to 0.0 C? 8. Temperature and Heat 4

5 THERMAL EXPANSION: AREA EXPANSION Since the length of an object changes with temperature, it follows that its area changes as well Consider a square piece of metal of length L, thus the initial area is L 2 If the temperature of the square is increased by T, the length of each side increases from L to L + L L + L = L + αl T As a result, the square has an increased area A A = (L + L) 2 = (L + αl T) 2 = L 2 + 2αL 2 T + α 2 L 2 T 2 Since α 2 T 2 is very small, it can be ignored Thus A L 2 + 2αL 2 T = A + 2αA T This result also applies to an area of any shape A circular disk with area πr 2 will also increase its area by 2αA T when subject to a temperature change T 8. Temperature and Heat 5

6 THERMAL EXPANSION: VOLUME EXPANSION If the length and area of an object increase when subjected to a temperature change T, it follows that the object s volume also increases Consider a cube of length L, and its volume V = L 3 Increasing the temperature results in increased volume V = (L + L) 3 = (L + αl T) 3 V = L 3 + 3αL 3 T + 3α 2 L 3 T 2 + α 3 L 3 T 3 Neglecting the smaller contributions as we did before: V L 3 + 3αL 3 T = V + 3αV T This expression is valid for any volume Volume expansion is described in the same way as linear expansion, but with a coefficient of volume expansion, β (K -1) V = βv T 3αV T, where β = 3α 8. Temperature and Heat 6

7 THERMAL EXPANSION: VOLUME EXPANSION - EXAMPLE A copper flask with a volume of 150cm 3 is filled to the brim with olive oil. If the temperature of the system is increased from 6 C to 31 C, how much oil spills fro m the flask? 8. Temperature and Heat 7

8 SPECIAL PROPERTIES OF WATER Water is a substance rich with unusual behaviour In solid form (ice) it is less dense than the liquid form, hence icebergs float The solids of most substances are denser than their liquids, hence when they freeze, their solids sink! The density of water changes over a wide range of temperatures Its density is a maximum at about 4 C, thus when y ou heat water from 0 C to 4 C, it shrinks 8. Temperature and Heat 8

9 HEAT CAPACITY Heat (Q) is the energy transferred due to temperature differences and its units are Joules It takes 4186J of heat to raise the temperature of 1kg of water by 1 C The heat required for a 1 C increase varies from one substance to another, e.g. it takes only 129J of heat to increase the temperature of lead by 1 C The heat required for a given increase in temperature is given by the heat capacity of a substance Heat capacity: C = Q/ T Joules/Kelvin (J/K) The heat capacity is viewed as the amount of heat necessary for a given temperature change An object with a large heat capacity (e.g. water) requires a large amount of heat for each increment in temperature Heat capacity is always positive, just like speed, so Q and T must have the same sign Q is positive if T is positive: heat added to the system Q is negative if T is negative: heat removed from the system 8. Temperature and Heat 9

10 SPECIFIC HEAT Heat capacity varies not only with the type of substance, but also with the mass of the substance Since it takes 4168J to increase the temperature of 1kg of water by 1 C, it takes twice that much to make the same temperature change in 2kg, and so on Thus a new quantity is defined the specific heat, c given by c = Q/m T (J/kg.K) Water has the largest specific heat, which means it can take in (or give off) large quantities of heat with little change in temperature If you take a pie out of the oven, the filling is much hotter than the crust because it has a high water content 8. Temperature and Heat 10

11 SPECIFIC HEAT PROBLEM SOLVING Suppose a block of mass m b, specific heat c b, and initial temperature T b is dropped into a calorimeter (a lightweight, insulated flask) containing water The water has mass m w, specific heat c w and initial temperature T w The aim is to find the final temperature of the block and water, assuming the calorimeter is light enough for it to be ignored, and that no heat is transferred from the calorimeter to its surroundings Remember that the final temperature of the block and water will be equal, and that the total energy of the system is conserved Thus the amount of energy lost by the block is equal to that gained by the water (or vice versa) Mathematically: Q b + Q w = 0 (meaning that the heat flow from the block is equal and opposite to the heat flow from the water) Rewriting Q in terms of specific heats and temperatures m b c b (T T b ) + m w c w (T T w ) = 0 Remember that T = T final T initial T = [m b c b T b + m w c w T w ]/[m b c b + m w c w ] 8. Temperature and Heat 11

12 SPECIFIC HEAT: EXAMPLE A 0.5kg block of metal with an initial temperature of 54.5 C is dropped into a container holding 1.1kg of water at 20.0 C, If the final temperature of the block water system is 21.4 C, what is the specific heat of the metal? Assume the container can be ignored and that no heat is exchanged with the surroundings. 8. Temperature and Heat 12

13 CONDUCTION Conduction is a form of heat exchange it is the flow of heat directly through a physical material From a microscopic point of view, holding one end of a metal rod with the other end in a fire, the high temperature of the fire causes the molecules at the far end of the rod to vibrate with an increased amplitude Neighbouring molecules start to vibrate as well, and this effect propagates through the rod, resulting in the macroscopic phenomenon of conduction A wooden rod in the same situation would behave differently the hot end would catch fire, whereas the other end would stay cool Conduction depends on the type of material involved Conductors conduct heat very well, whereas insulators do not 8. Temperature and Heat 13

14 HOW MUCH HEAT FLOWS AS A RESULT OF CONDUCTION Consider a rod length L and cross sectional area A, with one end at temperature T 1 and the other at T 2 > T 1 Experiments show that the amount of heat, Q, that flows through this rod: Increases in proportion to the rod s cross sectional area, A Increases in proportion to the temperature difference, T = T 2 T 1 Increases steadily with time, t Decreases with the length of the rod, L Combining these observations mathematically gives Q = ka( T/L)t The constant k is referred to as the thermal conductivity, and varies from material to material 8. Temperature and Heat 14

15 CONDUCTION: EXAMPLE Two 0.525m rods, one lead, the other copper, are connected between metal plates held at 2.0 C and 106 C. The rods have a square cross section, 1.5cm on a side. How much heat flows through the two rods in 1.0s? Assume no heat is exchanged between the rods and the surroundings, except at the ends. 8. Temperature and Heat 15

16 CONVECTION When heating a room with an electric heater, the heater s hot coils heat the air in their vicinity As this air warms up, it expands and thus becomes less dense Because of its lower density, the warm air rises, and is replaced by descending colder, denser air from above This sets up a circulating air flow that transports heat from the coils to the air in the room This type of heat exchange is called convection Convection occurs when a fluid is unevenly heated In convection, temperature differences result in a flow of fluid (hot air rising, cold air sinking) 8. Temperature and Heat 16

17 EXAMPLE OF CONVECTION During the day, the sun warms the land more rapidly than the water The rocky land has a lower specific heat than water The warm land heats the air above it which becomes less dense and rises Cooler air from over the water flows in to take its place, producing a sea breeze At night, the land cools of more rapidly than water because of its lower specific heat Now it is the air above the relatively warm water that rises and is replaced by cooler air from over the land, producing a land breeze 8. Temperature and Heat 17

18 RADIATION All objects give off energy as a result of radiation The energy radiated by an object is in the form of electromagnetic waves (visible light, infrared, UV) Unlike conduction and convection, radiation has no need for a physical material to mediate the energy transfer, and thus can travel through a vacuum The radiant energy from the Sun reaches Earth across 150 million kilometres of vacuum Since radiation includes visible light, it is possible to see the temperature of an object (i.e. glowing red hot) The radiated power, P, is the energy radiated per time by an object It is proportional to the surface area, A, over which the radiation occurs It also depends on the temperature of the object, specifically the fourth power of temperature T 4 (Kelvin) This behaviour leads to the Stefan-Boltzmann law P = eσat 4 (Watts) Stefan-Boltzmann constant: σ = W/(m 2.K 4 ) The emissivity, e, is a dimensionless number between 0 an 1 that indicates how effective the object is in radiating energy (1 means the object is a perfect radiator, such as a dark coloured object) Objects absorb radiation according to the same law by which they emit radiation If T is the temperature of an object, and T s the surrounding temperature, P net is the net radiated power P net = eσa(t 4 T s 4) 8. Temperature and Heat 18

19 NET RADIATED POWER If the object s temperature is greater than its surroundings, it radiates more energy than it absorbs and P net is positive If its temperature is lower than the surroundings, it absorbs more energy than it radiates and P net is negative When the object has the same temperature as its surroundings, it is in equilibrium and P net is zero Example: A person jumps into a lake of icy water. He has a surface area of 1.15m 2 and a surface temperature of 303K. Find the net radiated power from this person when he is inside where the temperature is 293K; and when outside where the temperature is 273K. Assume an emissivity of 0.9 for the person s skin. 8. Temperature and Heat 19