# Chapter 10 Temperature and Heat

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1 Chapter 10 Temperature and Heat GOALS When you have mastered the contents of this chapter, you will be able to achieve the following goals: Definitions Define each of the following terms, and use it an operational definition: temperature mechanical equivalent of heat thermometer heat capacity heat specific heat linear expansion latent heat of fusion volumetric expansion latent heat of vaporization calorie heat of combustion Calorimetry Solve problems in calorimetry. Gas Laws Solve problems using the gas laws involving the pressure, volume, and temperature of a confined gas. PREREQUISITES Before beginning this chapter you should have achieved the goals of Chapter 5, Energy, and Chapter 9, Transport Phenomena.

3 DEFINITIONS TEMPERATURE The relative hotness or coldness of an object. The human body contains many temperature sensors to inform you of the temperature of your body and its environment. THERMOMETER A device or transducer which can be used to measure the temperature of a system. The most common kinds of thermometer are mercury in glass and red-colored alcohol in glass. HEAT Thermal work, a process by which the internal energy of a system is changed. LINEAR EXPANSION Change in length accompanied by a change in temperature. Most materials increase in length when their temperatures are increased. VOLUMETRIC EXPANSION The change in volume accompanied by a change in temperature. Most substances expand when they are heated. Water at 4øC expands if it is heated and if it is cooled! CALORIE The mean amount of heat required to increase the temperature of one gram of water one degree Celsius. MECHANICAL EQUIVALENT OF HEAT The amount of mechanical energy which is equivalent to one unit of heat or thermal work. In the SI units joules is equivalent to 1 calorie. Your (Fuller) 3 textbook weighs about 13 Newtons. If you lift it about 32 cm you will have done about 4.19 joules of work. That amount of work would raise the temperature of 1 cubic centimeter of water one degree Celsius. HEAT CAPACITY Product of the mass of a body in grams and specific heat; i.e., amount of heat required to raise the temperature of the body one degree Celsius. SPECIFIC HEAT Heat energy required to raise the temperature of a mass of material one Celsius degree. Compared to many other substances, See Table 10.3, water has a large specific heat. Water is a good substance to use for the storage of thermal energy. LATENT HEAT OF FUSION Heat required to change unit mass of material from the solid state to the liquid at the melting point. Consider how high you would have to lift your copy of (Fuller) 3 to do the amount of work equivalent to the latent heat of fusion of one cubic centimeter of ice at 0 ø C. More than 25 meters!

4 LATENT HEAT OF VAPORIZATION Heat required to convert unit mass of material from liquid state to vapor state at the boiling point temperature. Guess how high you would have to lift your copy of (Fuller) 3 to do an amount of work equivalent to the amount of energy required to convert one cubic centimeter of hot water at 100øC to steam at 100øC. About 173 meters! That's much higher than the state capitol building in Lincoln, Nebraska (which is about 130 meters tall)! HEAT OF COMBUSTION Energy produced per unit measure of quantity of material for complete oxidation. Which would you rather oxidize, spinach or beef steak? You will get about 20 times more energy from the steak as from an equal mass of spinach. ANSWERS TO QUESTIONS FOUND IN THE TEXT SECTION 10.2 Temperature Absolute zero in Celsius is ø C. In Fahrenheit, absolute zero is ø F. SECTION 10.4 Volumetric Expansion Let us consider a cube of side L 0, then its volume is V 0 at a temperature T 0. Now let us heat the cube to a new temperature ΔT degrees hotter than T 0. Now we calculate the new volume V in terms of the volumetric coefficient of expansion β, V = V 0 + ΔV = V 0 +βv 0 ΔT = V 0 (1 +βδt) (1) But we can use the linear measurement to calculate this same volume. V = L 3 = (L 0 +αl 0 ΔT) 3 = L 0 3 (1 + αδt) 3 = V 0 (1 + αδt) 3 = V 0 (1 + 3αΔT + 3α2ΔT 2 +α3δt 3 ) If α is much smaller than 1; then α 2 << α; so neglect the α 2 and α 3 terms: V V 0 (1 + 3αΔT) (2) By comparing equations (1) and (2) we see β 3α SECTION 10.8 Heat of Combustion At this date the cost of butter is about \$.80 per pound and gasoline costs about \$.65 per gallon. Energy from butter = 716 kcal/100gm x 454 gm/1 lb. x 1 lb./\$.80 = 4060 kcal per dollar Energy from gasoline = 1150 kcal/100 gm x 4546 cm 3 /gallon x 1 gallon/\$0.65 x 0.68 gm/1 cm 3 = kcal per dollar Butter costs 13 times as much as gasoline per unit of energy. EXAMPLES CALORIMETRY 1. A nurse withdraws 50 cc's of blood from the arm of a patient. He wants to keep the sample for a while, so he puts it in a refrigerator and cools the sample to 2 ø C. How much heat in calories must be removed from the blood to do this? Assume for this problem that 50 cc's of blood has a mass of 52 gm; assume that the specific heat of blood is the same as the specific heat of water; assume that the body temperature of the patient is 98.6 ø F or 37 ø C. Be sure to give the correct units.

5 What Data Are Given? Mass of blood = 52 gm; initial temperature of blood = 37 ø C; final temperature of blood = 2 ø C, specific heat of blood = 10 3 calories per kg. What Data Are Implied? None, it is all given. What Physics Principles Are Involved? The basic Calorimetry equation (10.9) is all that is needed. What Equations Are to be Used? ΔQ = mcδt (10.9) Algebraic Solution Heat Removed = ΔQ = mcδt Numerical Solution ΔQ = (52 gm)(1 cal/gm)(2 ø C - 37 ø ) = -1.8 x 10 3 calories Thinking About the Answer The change in heat, ΔQ, is negative because heat is lost from the system of blood. 2. Heat is required to change ice (solid) to water (liquid) without changing its temperature. In fact, it requires 80 calories of heat to change 1 gram of ice at 0 ø C to 1 gm of water at 0 ø C. Furthermore, heat is required to change water (liquid) to steam (gas) without changing its temperature. In fact, under standard conditions it requires 540 calories of heat to change 1 gm. of water at 100 ø C to 1 gm. of steam at 100 ø C. (a) How much total heat is required to warm and convert a 1 gm piece of ice at 0 ø C to steam at 100 ø C under standard conditions? (b) What percentage of the total heat is used to change the 100 ø C water to 100 ø C steam? What Data Are Given? 1 gram of ice exists at 0 ø C. What Data Are Implied? During the heating process, no heat will be lost from the ice - water - steam system to its surroundings. What Physics Principles Are Involved? The basic concepts of Calorimetry (Equation 10.9) and changes in state (Section 10.7) are needed. What Equations Are to be Used? Heating the water = ΔQ = mcδt (10.9) Melting the ice = ΔQ f = L f m (3) Vaporizing the water = ΔQ v = L v m (4) Algebraic Solution (a) total heat = ΔQ f + ΔQ +ΔQ v = ml f + mlδt + ml v (5) (b) percentage for vaporization = ml v /total heat (6) Numerical Solutions (a) Q total = (1)(80) + (1)(1)(100-0) + (1)(540) = 720 calories (b) Percentage = 540/720 = 75% Thinking About the Answer Notice how much energy is required to convert the liquid water to the gaseous water, 75% of all the energy!

6 3. About how many hamburgers would a 91 kg (200 lb) man have to eat to enable him to climb to the top of the Nebraska capitol building, about 131 meters high? An average hamburger has a fuel value of about 250 kilocalories. The human body has a 10% efficiency; i.e., 90% of the fuel value of food is used to maintain the body, only 10% is available for work. What Data Are Given? Mass = 91 kg; height to be lifted = 131 meters, efficiency = 10%, heat of combustion = 250 kcal/hamburger. What Data Are Implied? The energy intake must be 10 times the amount used because the useful work is only 1/10 of the total energy input. Assume g = 9.8 m/s 2. It is assumed that the total fuel value of the hamburgers is used by the human body. What Physics Principles Are Involved? The mechanical equivalent of heat is what is needed for this problem. The potential energy needed or the work done, to climb to the top of the Nebraska capitol building. What Equations Are to be Used? Work done in climbing = ΔP.E. = mgh Number of hamburgers = work done in kilocalories / (efficiency)250kcal/hamburger Algebraic Solution N = W = mgh [W(1 kcal/4186 J)/250 kcal/lb.]/(efficiency) Numerical Solution W = (91 kg)(9.8 m/s 2 )(131 m) = 1.2 x 10 5 J W = 28 kcal Energy needed = W/efficiency = 28/.1 = 280 kcal. N = 280 kcal / 250 kcal/lb. = 1.1 hamburgers. During his climb up to the top he would use up about the energy of one hamburger for climbing and for maintaining his body conditions. Thinking About the Answer If you assume his other food is used to maintain his body conditions and he ate an extra hamburger, in order to keep from gaining weight he would have had to climb to the top of the capitol building 9 times. This shows why dieting rather than exercise is the easiest way to maintain, or reduce, body weight. GAS LAWS 4. A weather balloon contains 280 m 3 of helium gas at sea level where the atmospheric pressure is 1.01 x 10 5 N/m 2 and the temperature is 27 ø C. Calculate the volume of the balloon when it reaches an altitude of 16 km where the pressure is 1.00 x 10 4 N/m 2 and the temperature is -55 ø C. What Data Are Given? V 1 = 280 m 3 ; P 1 = 1.01 x 10 5 N/m 2 ; T 1 = 27 ø C or 300 ø K; P 2 = 1.00 x 10 4 N/m 2, and T 2 = - 55 ø C or 218 ø K. What Data Are Implied? That the balloon skin applies no extra pressure to the gas on the inside of the balloon, so the gas laws can be applied directly using the given data. It is assumed helium is an ideal gas. What Physics Principles Are Involved? The combined laws of Boyle's and Charles' for ideal gases can be used.

7 What Equations Are to be Used? P 1 V 1 /T 1 = P 2 V 2 /T 2 (10.15) Algebraic Solution V 2 = (T 2 /T 1 )(P 2 /P 1 ) V 1 Numerical Solution V 2 = (218 ø K/300 ø K) x (1.01 x 10 5 /1.00 x 10 4 ) (280 m 3 ) V 2 = 2060 m 3 Thinking About the Answer The effect of the greatly reduced pressure to allow the balloon to expand is much more important than the reduced temperature that causes the gas in the balloon to have a reduced volume. The reason for this, of course, is the fact that absolute zero is many degrees below the range of temperatures occurring in this problem. Suppose absolute zero were only -60 ø C instead of -273 ø C. How would the results of this problem be changed?

8 SPECIAL PRACTICE PROBLEM FOR FUN 4. A well-known secret fact about Robinson Crusoe is that he was an amateur entomologist. While shipwrecked on his island every day at noon (sun directly overhead) he counted the number of ants coming out of the largest anthill in about one minute (70 beats of his heart). He noticed that on hot days the ants were more active than on cold days, so he constructed his own Robinson Crusoe temperature scale. He assumed there was a linear relationship between the number of ants he counted and the temperature. He made his own simple temperature scale, zero ants was 0 ø RC and 100 ants was 100 ø RC. After he was rescued he desired to calibrate his ant hill from known meteorological data and to fill in the missing numbers in his daily log. Fill in the missing numbers in the table below: ANSWERS: The relationship between the two temperature scales is given by T(RC) = 2.5 T( ø F) So 86 ø F = 65 ø RC; 92 ø F = 80 ø RC; 72 ø F = 30 ø RC. Do you think this is a creepy problem?

9 PRACTICE TEST 1. Find the Celsius temperatures corresponding to the following common Fahrenheit temperatures. a. Room temperature (68 ø F) b. Human body temperature (98.6 ø F) c. Cold winter temperature (-4 ø F) d. Ice melts (32 ø F) e. Water boils (212 ø F) 2. An automobile tire at 32 ø F has a gauge pressure of 24 lb/in 2 (then its absolute pressure is 2.6 x 10 5 N/m 2 ). Immediately after running at high speed on the interstate highway, the tire pressure is measured to be 32 lb/in 2 (its absolute pressure is 3.2 x 10 5 N/m 2 ). What is the temperature of the tire, assuming the volume of the tire remained constant? 3. On a hot summer day (40 ø C) a physics student fills the 72 liter gasoline tank of her car with cool gasoline (10 ø C) from a self-service pump. Why will the gasoline overflow the tank? How much gasoline will flow out of the tank? (Coefficient of volumetric expansion of gasoline = 0.90 x 10-3 / ø C) 4. A 5 kilogram lead sphere falls from a 100 meter high building and lands on a hard concrete sidewalk. If all the heat energy created at impact is assumed to be retained by the sphere, answer the following questions. a. How much heat is generated at the impact? b. What is the final temperature of the lead if its initial temperature was 22 ø C? (Specific heat of lead =.13 J/gm ø C). ANSWERS: 1. a. 20 ø C b. 37 ø C c. -20 ø C d. 0 ø C e. 100øC ø C 3. Increase in the temperature of the gasoline causes volumetric expansion; 1.9 liters 4. a. 5,000 J b ø C

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