Understanding the Difference Between Temperature and Heat

1. Thermology Chapter 1

2. Thermology regards phenomena related to heat exchanges in bodies and their temperature. The terms hot, cold, heat and temperature are commonly used but the last two are often misunderstood. It is better, from the very beginning, to point out that the physical quantities of temperature and heat are different and that we often confuse or identify them . To understand the difference, I will give you an example. If we pour two different quantities of water into 2 equal containers “A” and “B”, we will notice the following: In the container “A” the level of the water is h, while the level of the water in the container “B” is h’. As there is more water in the container A than in B, the level of the water (h), wil be higher in A than in B (h’). This means that the level of the water , h, indicates the quantity of water contained in the container, and the level h is higher when the quantity of water is greater. Now, if the container A is not equal to B, because they haven’t the same shape or capacity, and we pour the same quantity of water, what happens? The liquid will appear as follows The level of liquid isn’t the same in the two containers.

3. This time can we say that there is more water in the container A than in B? No, we can’t because the 2 containers have not the same shape or capacity . The temperature indicates the thermic conditions of a body, that is the quantity of heat a body possesses ( that is, how hot a body is). If two bodies with the same mass, have the same temperature, they possess the same heat; but if they have not the same mass, if the first body has a higher temperature than the second, we cannot say that it has a major quantity of heat (as in the case of the two different shaped containers). The particles of matter (atoms or aggregates) are in constant motion. In solid they vibrate (wiggle) around a fixed point, while in fluids (liquids and gases) they move in a disorderly way along long zig zag tracks, due to the bangs with other particles or the walls of the container. Each particle, in whichever state of aggregation , possesses Kinetic Energy : The heatof a bodyis the total kineticenergypossessedbyallitsparticles . Heatistherefore a formofenergy, alsocalledthermalenergy. Obviously, if the body has very few particles, that is a small mass, it will have very little heat or heat energy. When we add the kinetic energy of all the particles and divide the result by the number of these particles, we obtain the average kinetic energy of each particle . K = Boltzmann’s constant = 1,38 ∙10-23 J/K The temperatureof a body is a degreeof the termiccondition. and depends on the averagekineticenergyofitsparticles. If the temperature increases, the chaotic movement of the body particles increases.

4. Therefore, if the body has a high temperature, the particles will move more chaotically, with a major kinetic energy average. but if the body has a small mass, that is few particles, the total quantity of heat possessed by the body is low. This is the case of a lit candle which has a high temperature, but produces very little heat, not enough to heat a room in the winter period. In the same way if some water in a spoon is heated by a flame, the temperature of the water will be high (in this case, it would burn our lips), but the quantity of heat possessed is low, surely inferior to the water contained in a bathtub full of warm water. The warm water of the tub possesses a low kinetic energy average of its particles, because the temperature is low, but as it has a major quantity of water it will have a major quantity of heat. The water in a radiator is at a relatively low temperature (40°C) but the quantity of heat is high and can heat, in the winter period, the room in which the radiator is located. Thus, a high temperature of a body, may not have a high quantity of heat and viceversa. Temperature and heat are different physical quantities, even if there is an interconnection

5. Another example A B n=10 n=10 Let’s consider two towns A and B with the same number of inhabitants n=10 € 100 € 50 Each inhabitant living in town A earns 100 € a month while each inhabitant in town B earns only 50 €. Now let’s find out: - which of the two inhabitants, A or B, is the richer? The answer is, without doubt, the inhabitant A who earns an average of 100 € a month, while the inhabitant B earns an average of 50 € a month. - Which is the richest town A or B? The answer is town A, in fact Total richness of A =10*100 € = 1000€. Total richness of B=10*50 € = 500€. Now let’s suppose that the two towns don’t have the same number of inhabitants, but town A has n=10, while B has n=100 and that the inhabitant in town A, as before, earns €100 while the inhabitant B earns €50. Now let’s find out: - which of the two inhabitants is the richer? The answer is obviously the same before: the inhabitant A who earns €100 a month. - which is the richest town A or B? the answer, this time, is town B. Total richness of A =10*100 € = 1000€. Total richness of B=100*50 € = 5000€. Heat indicates the wealth of the two towns, that is the total kinetic energy of the molecules of the body, while the temperature indicates the average richness of the inhabitants, that is, it is related to the average kinetic energy of the body molecules.

6. In conclusion (or so), we can say that…): If we say that a body has a lot of heat, we mean that the total kinetic energy of all its particles is very high. If we say that its temperature is high, we mean that the average kinetic energy of its particles is high (and the body may not, in this case, have or produce much heat). Often, when we pour water into a bathtub and then put our hand in we usually say, according to the case, that the water is either hot or cold. These feelings of heat and/or coldness are (subjective) personal and relative, indeed a rougher hand may have different feelings. Temperature is measured in another way objectively! Please note that it is incorrect to say that the water is hot and therefore has a lot of heat - you must say that the body possesses a high temperature. So, when we say that a body is cold meaning that it has very little heat is incorrect; the right way of expressing this condition is that the body has a low temperature. The terms “hot” and “cold” refer to heat and not to the temperature, thus, it is correct to say that a body has a high temperature (instead of “it’s hot”) or a low temperature (instead of “it’s cold”).

7. Let’s consider two communicating containers A and B, with a different quantity of water that will reach two different levels (picture 1). What will happen? As we know, the water in A with a higher level, h1, will move into the container B, which has a lower level, h2. This movement will continue until a state of equilibrium is obtained, that is, the level of the water in both containers is the same h (picture 2). What will happen, if we place two bodies A and B at different temperatures t1 and t2 (whith t2 > t1) and different heat energy (possibly A (less than B) next to each other? t1 t2 hot object cold object There will be a flow of heat (that is not liquid) from the body with a higher temperature to the body at a lower temperature, until the two bodies reach the same final temperature (thermal equilibrium). ). Such as condition is colled: thermalequilibrium

8. Two bodies at different temperatures, when put in contact, after a while… t1 > t2 ...will reach the same temperature thermal equilibrium Two bodies in thermal equilibrium have the same temperature. In conclusion: thermal equilibrium is achieved when two objects at different temperatures are placed next to each other, the object with more temperature will give heat to the other object with less temperature , until the two objects reach the same temperature. Thermal equilibrium is that condition in which two bodies, at different temperatures, place near each other, reach the same temperature.

9. The temperature, indicated with the symbol t, is a scalar physical quantity, which is measured in degrees. The thermometeris the deviceusedtomeasuretemperature. A thermometer has a bulb that contains thermometric liquid (mercury), connected to a cylindrical capillary tube that is closed at the other end. 100 Calibration of a thermometer Let’s put a thermometer into a stirred mixture of water and ice: the mercury will reach a level that is conventionally indicated with ZERO. Now let’s put the thermometer into boiling water: the mercury will expand and reach, according to convention, a level indicated with 100. 0 0 Finally , let’s divide the distance between 0 and 100 into 100 equal parts . In this way we have calibrated the thermometer . Thermometer scales The above thermometer uses a centigrade scale , that is divided into 100 equal parts that represent Celsius centigrade degrees, indicated with the symbol “C” . This scale, also called Celsius scale, takes its name from the Swedish scientist who introduced it . We use this thermometric scale .

10. The French use another scale between 0 and 80 (Reaumor Scale), indicated with R, while the English use the Fahrenheit scale (that goes from 32 to 212) indicated with F. The International Scientific Community, uses in SI (International System) the Kelvin Scale (symbol K), named for the British physicist Lord William Thomson Kelvin (1824-1907). The temperature of a Kelvin scale is called absolute temperature. In this scale -273°C corresponds to 0 K (called absolute zero, indicating a theoretical unattainable temperature ( as everything disappears, the bodies do not exist); 0°C corresponds to 273 K . Thus T (absolute temperature) = t (temperature in Celsius degrees +273). T (k)= t (°C)+273 212 373 80 100 It is possible to convert one scale into another by using the formulas: C: 100 = R : 80 (from Celsius to Reaumur) C : 100 = ( F-32 ) : 180 (from Celsius to Fahrenheit ) 180 100 100 80 0 273 32 0 T = t + 273 from Celsius to Kelvin 0 -273 -273 -273 Esempio 1) If someone has a 37 °C temperature, .which is his/her temperature in K, R and F scale ? 1) T = t + 273 = 37 + 273 = 310 K C: 100 = R : 80 2) 37: 100 = R : 80 tR = 3780 /100 = 29,6° R tF - 32 = 37 180 / 100 = 66,6 tF = 66,6 + 32 = 98,6° F 37 : 100 = ( tF- 32 ) : 180 ∙ 3) C : 100 = ( F- 32 ) : 180

11. Different scales are used to measure temperature: the Fahrenheit, Celsius and Kelvin scales are the most common in use today.

12. Another example: - A summer day in America has a temperature of 120° F. What’s the correspondent temperature in centigrades and according to the Reamur and Kelvin scales? 1) C : 100 = ( F-32 ) : 180 tC : 100 = (102 – 32) : 180 tC : 100 = 70 : 180 tC = 100 * 70 / 180 = 38,8° C tR = 38,8°C * 80 / 100 = 31,4° R 2) T = t + 273 = 38,8 + 273 = 311,8 K Exercises 1) If the temperature is 24° C, express it in R and F degrees and in K ! 2) If the temperature of a body is 152° F, what is its temperature in C and R degrees and in K? PHISICS AROUND US The infrared thermometer is commonly used. It determines the temperature indicating the infrared radiation emitted by the body. Please note: mercury was the first thermometrical liquid used, but today alchol or galinstan ( a liquid mixture of gallium, indium and tin at environment temperature= ambient/room temerature) is used.

13. Thermal expansion is the tendency of matter to change in length (or area or volume) in response to a change in temperature. Linear thermalexpansion Now let’s heat a metal bar/rod that has a length Lo from the initial temperature T1( for example 10°C) to the final temperature T2 (for example 110°). We will observe that after a while it will expand, that is, its length increases. L0 Indicates the variation of temperature Lt L0 Lt In this case we speak about the law of linear thermal expansion: Let’s state the law of linear thermal expansion: The lengthening of the body is directly proportional to the initial length, to the variation of temperature and to the type of material used in the bar. Each material/substance expands differently and therefore has its own linear expansion coefficient: We know that :

14. As each substance is different, it expands in a different way and therefore has its own coefficient of linear expansion . n the IS it is measured in K-1. Due to linear expansion, it is necessary to consider (and foresee) this phenomenon when constructing buildings. When railway tracks are built, for example, thermal expansion is taken into consideration and a space is left in between tracks. (See picture). The pipes of a gas pipeline undergo consistent temperature changes. To absorb length variations we insert arches with an omega shape that warp in a non-permanent way, along rectilinear sections. The noise we hear when we are on the train : tu-tum-tu-tum is caused by the wheel of the train that passes over the tracks joints. The length of a bridge section changes from summer to winter. Joints are added to connect sections and allow transition. When cars pass over joins in the winter period, they make more noise as the spaces in between are larger. The opposite of thermal expansion is contraction and occurs when the temperature decreases , thus the final length diminishes!

15. lt = l0 ( 1 + Exercise A metal bar is 2.5 metres long and has a temperature of 0 °C. Determine its length at the temperature of 65 °C. What is the difference in length?

16. Volume thermalexpansion Solids do not expand in one direction but in its three dimensions of space (width, lenght and height). In this case we are talking about cubic or volumetric thermal expansion. A solid with a volume V0 undergoing a variation of temperature will present a change in volume: The law of volumetric thermal expansion of a solid. Where is the coefficient of the volumetric expansion that depends on the material of the solid is measured in K-1 o °C-1 As for the law of linear thermal expansion, the law of volumetric expansion can be expressed as follows: where: The law of volumetric expansion is also applied to liquids and gases. An example of volumetric expansion is given by a metal ball made to fit and pass through a metal ring. If the ball is heated, its volume increases and does not pass through the ring.

17. Thermal expansion is the tendency of matter to change in volume in response to a change in temperature. The length, area or volume variations ( ) of a solid bar are related to changes in its temperature (∆T), to its original length, area or volume ( ) and to its expansion coefficients ( ,or ). These expansion coefficients are specific to the material the bar is made of. The expansion coefficient is measured in K–1.

18. The coefficient of linear expansion and the coefficient of volume expansion are roughly related by Pyrex – glass = 3 These expansion coefficients are specific to the material the bar is made of.

19. Unusual behaviour of water. Water in the liquid state presents an anomaly , indeed most substances , by diminishing the temperature, diminish their volume and therefore increase their density : This also occurs with water, unless it reaches a temperature of 4° C . Between 4° C and 0° C , the volume of water does not decrease , but increase , while its density diminishes . This property of water has important effects on the life on Earth . During the winter period the temperature of the surface layer of the sea and lakes diminishes : the water becomes denser and moves down to the bottom , while the hotter water of the bottom moves to the surface and becomes colder . This cycle ends when the water reaches the temperature of 4° C ; in fact a further cooling of the temperature ( with a consequential increase of volume ) makes the water less dense and remains on the surface until it changes into ice. In the underlying layers fish and others forms of life go on living in the water in a liquid state . At the temperature of 4°C density undergoes a change Cyclic movement that ceases at 4° C

20. Ice floats; unlike most other substances, water has a lower density in the solid state than in the liquid state.

21. We have seen that a variation of temperature causes a change of volume in solids and liquids,while in the case of gases it will not only change their volume but also their pressure. Now let’s consider the behaviour of a gas, and for the sake of simplicity, of a perfect gas.. By perfect gas or ideal gas (that does not actually exist), we intend a sample similar to a real gas when: - the density is very low (as well as pressure); - the temperature (high) is far from the temperature of liquefaction(change of state from gas to liquid). Some real gases, such as hydrogen, helium and the air (at ambiente/room temperature) c an be considered perfect when in normal temperature and pressure conditions. • To study a gas with a given mass m (Kg) we can put a quantity inside a scaled watertight cylindrical container, with a moving piston and measure: • the volume V (in m3), the pressure p (in Pa) , the temperature t (in °C ) o T (in K) weights These three quantities : pressure, volume and temperature define the state of gases and are therefore called quantities or variables of state. If even just one of these three quantities varies, the gas passes from the initial state "A" to another final "B", and it is said that the gas has undergone a transformation or change of state. piston These three quantities : pressure, volume and temperature are not independent but related; the change of one varies the others even in a complicated way, however the following strategy is used : for a given quantity of gas; one of the quantities is kept constant and the relationship between the others is observed. .

22. During the following experiment,the gas temperature is kept constant. Let's put gas into a scaled cylinder which has,at the very top,a piston free to move without friction, the weight of this gas is not considered and no external pressure acts on it while the temperature is kept constant and invariable in this trasformation. free piston Let's put a weight on the piston, and when the piston stops, we can determine the volume value. As the section S of the piston is constant, it is necessary to increase the force F, cylinder that is, the number of weights on the piston, to increase the pressure p: with their relative units of measure We can observe, see table, that the volume gets smaller and smaller as we increases the number of weights on the piston and therefore the pressure N.B. A transformation that maintains a constant temperature is called isothermal transformation.

23. We can say that the variations of the volume of a gas is in relation to pressure in: Boyle’s law (transformation at a constant temperature) In an isothermal transformation pressure and volume magnitudes are inversely (if the first increases, the second will decrease, and viceversa) . Boyle’s law : the pressure is inversely proportional to the volume. From the previous table, if we multiply, for each pressure change, the corresponding value of the volume, we have: So we have: the product of pressure and volume is always constant, in our case this volume is= 150. p∙V = cost It is another way of expressing Boyle’s Law Thus, generally speaking, we can say that: where 1 indicates the initial state of the gas and 2 it’s the final state p1∙V1 = p2∙V2 or also : isoterma isotherm If we represent Boyle’sLaw, it is in a Cartesian plane where the values of the volume V are put on the abscissae and the values of pressure P on the ordinates (the Clapeyron plane), we obtain: a branch of the hyperbole, which is a characteristic of all inverse proportionalities. branch of the hyperbole N.B. Please note: by state A of a gas we mean the state in which the gas is and characterized by specific values of p, V and T.

24. We can repeat the experiment that has always a constant temperature, but by examining different temperatures we obtain different isotherms, as in the given picture below: The isotherms which are further away from the origin correspond to transformations that occur at higher temperatures: the transformation 1 occurs at a temperature T1, which is less than T2 related to the transformation 2, which is in turn less than T3 related to the transformation 3. Example: If in an isothermal transformation the initial pressure of the gas is 400 Pa, its initial volume is 0,5 m3 and the final pressure is 200 Pa, what is its final volume? In symbols: p1= 400 Pa V1= 0,5 m3 p2= 200 Pa V2= ? p1∙V1 = p2∙V2 Exercise: If in an isothermal transformation the initial pressure of the gas is 800 Pa, its initial volume is 0,5 m3 and its final volume is 2m3, what will its final pressure be?

25. Gay Lussac’s first law: heating at a constantpressure. Let’s consider a given mass of gas enclosed in a container with a constant external pressure (what means without varying the number of weights on the piston) . We will observe that by increasing the temperature, provided by a gas stove, the volume will increase ( so the piston, free to move goes up, according to the following experimental law, known as Gay - Lussac’s first law: Where Vt represents the final volume of the gas at a generic temperature t, Vo is the initial volume of the gas at the temperature t0 = 0° . Finally, the thermal expansion coefficient is constant for all gases and is: that is the increase of gas volume is proportional to the increase of temperature and is always the same for any kind of gas: for each degree 1/273 initial value (Vo) isobar Increase of volume Linear graphical representation of Gay-Lussac’s first law, on the cartesian plane. Burner is off Burner is on A transformation that has a constant pressure is called : isobar.

26. Example What is the final volume of a gas, knowing that its initial volume is equal to 100 cm3, its initial temperature is 0 °C and its final temperature is equal to 50°C? In symbols: V = ? Vo = 100 cm3 =100•10-6 m3 = 10-4 m3 to= 0°C t = 50°C

27. Let’s rewrite Gay–Lussac’s first law replacing the temperature t expressed in °C with the temperature T (absolute temperature) expressed in K: Let’s suppose that the initial temperature of the gas is: t0 = 0 , so: In fact that expresses , at a constant pressure, the direct proportionality between the volume V (in m3) and the temperature (in Kelvin). That can also be witten: If we graph (V, T) how the volume changes in relation to the absolute temperature , we have: Generally speaking, between two states A and B of a gas, we can write: Isobar P = const or also: p = constant

28. Gay-Lussac’s second law : heating at a constant volume. Let’s consider a given mass of gas enclosed in a container where the volume is constant, blocking the moving piston of the cylinder, and let’s increase the temperature of the gas by providing heat . We will notice that an increase of temperature causes an increase in volume , according to the following relation, known as Gay-Lussac’s second law. Where pt represents the final pressure of the gas at a generic temperature t, po is the initial volume of the gas at the temperature t0 = 0°. The thermal expansion coefficient is constant for all gases and is : that is, the increase of gas pressure is proportional to the increase of temperature and is always the same for any kind of gas: for each degree1/273 initial value (po). Isochore: V=const Linear graphical representation of Gay-Lussac’s second law, on the cartesian plane. Burner is off Burner is on A transformation that has a constant volume is called : isochoric or isovolumic.

29. Example: What’s the final pressure of a gas, knowing that the initial pressure is 0,1 Pa (at 0° C), the initial temperature is 0°C and the final temperature is 100°C. In symbols: p = ? po = 0,1 Pa to= 0°C t = 100°C Absolute zero If we analyse Gay-hussac’s laws (p= p + αt and V= V + αt) we can see that the line intersects the axis of the temperature at -273 °C. This means that the pressure and the volume would be zero at that temperature; now a gas being formed of molecules in motion, can never have a volume or pressure equal to zero, because it would mean that nothing exists. The temperature -273°C is defined as absolute zero and represents the zero of the Kelvin scale, known as the scale of absolute temperatures.

30. Let’s rewrite Gay–Lussac’s second law replacing the temperature t expressed in °C with the temperature T (absolute temperature) expressed in K: Let’s suppose that the initial temperature of the gas is t0 = 0 , so: In fact that expresses , at a constant volume, the direct proportionality between the pressure P (in pa) and the temperature (in Kelvin). che può essere anche scritta: If we graph (p, T) how the Volume changes in relation to the absolute temperature, we have: Generally speaking, between two states A and B of a gas, we can write: Isochore: V=const or also:

31. WehavealreadyrepresentedBoyle’s lawon the Clapeyronplane, thatis a Cartesianplane (orientedaxesmutuallyorthogonal), where the volume (V ) values are indicated on the abscissas (x) and the pressure P on the ordinates (y ). Isotherm Transformation at a constant temperature branch of the hyperbole Nowlet’s representGay-Lussac’s first and secondlaw on the Clapeyronplane; wehave: Isobar Trasformation at a constant volume Trasformation at a constantpressure Isochore

32. • At constant temperature the product of the volume (V) occupied by a gas and its pressure (p) is always constant → Boyle’s law: • The volume occupied by a gas at constant pressure is directly proportional to its absolute temperature (T) → Gay-Lussac’s first law: • In a transformation at constant volume the pressure of a gas is directly proportional to its absolute temperature (T) → Gay-Lussac’s second law:

33. A perfect gas or ideal gas is a theoretical gas that does not actually exist; it is a simplified representation of real gas that gives us the possibility to state simple laws, to a good approximation, consistent with reality. It is possible to demonstrate that the three laws of gases previously examined, can be summed up by a single equation ,called equation of state of ideal gases: where : n indicates the number of moles 1 mole = Avogadro’s number of molecules = 6,022 •1023 mol-1 R = universal constant of gases = 8,31 J/mol• K W can obtain the three laws from the equation of state of ideal gas, thus: If T is constant, as n ed R are also constant, we have : Boyle’s law constant•T If P is constant, we have: Gay-Lussac’s first law constant•T If V is constant, wehave: Gay-Lussac’s second law

34. An ideal gas is a hypothetical gas that satisfies all the gas laws: Boyle’s law and both the Gay-Lussac laws: equationof state The equation of state summarises the three gas laws and it is possible to derive Boyle’s law and the Gay-Lussac laws from it as special cases; as shown in the table below.

35. If body A is in thermal equilibrium with body B and body B is in thermal equilibrium with body C, then body A and body C are in thermal equilibrium with each other. example: if a thermometer in a room records a temperature of 20 °C, all the other objects in the room, after a certain time, will reach a temperature of 20 °C