Chapter 10. Thermal Physics. Thermodynamic Quantities: Volume V and Mass Density ρ Pressure P Temperature T: Zeroth Law of Thermodynamics

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1 Chapter 10 Thermal Physics Thermodynamic Quantities: Volume V and Mass Density ρ Pressure P Temperature T: Zeroth Law of Thermodynamics Temperature Scales Thermal Expansion of Solids and Liquids Ideal Gas Macroscopic Description The Kinetic Theory

2 What is Thermal Physics? Thermal Physics is the study of system interactions involving transfer of energy as heat or/and work leading to the variation of the thermodynamics state of the systems Ex: Say that the moving box discussed on one of the previous slides is metallic: the heat released due to friction is absorbed by the material increasing the temperature, which will consequently determine an increase in volume and surface melting: these processes are thermodynamic, only in complementary relationship with the mechanical motion of the box Thermal Physics approaches its area of study at various scales macroscopic and microscopic, as well as via statistical mechanics which applies probability theory to large ensembles of particles to model macroscopic properties based on microscopic mechanical, electromagnetic and quantum mechanical characteristics of the atomic and molecular constituents of the system The thermodynamic state of a simple system is given by a relationship between several physical quantities: 1. Volume, V: describes the space filled by a certain amount of substance 2. Pressure, P: describes the force per unit area exerted in a material 3. Temperature, T: is a conventional parameter describing a thermal state In a thermodynamic process resulting in a change in the thermodynamic state, all or only two of these parameters may change in a correlated fashion

3 1. Volume and Mass Density The volume mass density ρ describes how a substance of mass m is distributed in the respective volume V: if the mass is distributed uniformly The concept of density can be extended to a mass distributed in plane within area A, or along a line of length L: 2. Pressure superficial mass density: linear mass density: ρ = m V σ = m A λ= m L Def: The pressure P associated with a force F exerted uniformly across a surface A is P F A 2 2 = newtons meters = N m = Pascal= Pa When a force is exerted, its effects will be different depending on how it is distributed on the surface Ex: Pushing a balloon against a nail is not a good idea, since the reaction to your push will be acted on a small surface. However, pressing the balloon against many nails is fine since the same force is distributed on a large surface. The difference between the two cases is not in force but in pressure

4 Exercise: pressure in a stationary liquid In order to operate with the concept, let s see how we can calculate the pressure at a depth h under the surface of a liquid of uniform density ρ. Consider for instance, a calm volume of sea water of density 1030 kg/m 3. For start, consider an arbitrary surface of are A at depth h. a) What force exerts a pressure onto the surface A? How can this force be expressed in terms of given quantities? b) Hence, what is the pressure at depth h in terms of given quantities?

5 Temperature: Zeroth Law of Thermodynamics Qualitatively, temperature T is a measure of how cold or warm is a certain substance, associated with a subjective perception of its content of thermal energy In order to conceptualize temperature we must introduce the definitions: 1. Two objects are called in thermal contact if they can exchange heat 2. If two objects in thermal contact do not exchange energy, they are called in thermal equilibrium Then, temperature can be defined using The Zeroth Law of Thermodynamics: If objects A and B are separately in thermal equilibrium with a third object, C, then A and B are in thermal equilibrium with each other Hence, the temperature is the property that determines whether or not an object is in thermal equilibrium with other objects, giving consistency to the zeroth law: two systems are in thermal equilibrium if they have the same temperature Quantitatively, the measurement of temperature makes necessary an instrument called a thermometer calibrated using a conventional scale of temperature

6 Temperature: Thermometers Thermometers are used to measure the temperature of an object or a system, making use of physical properties that change with temperature Here are some temperature dependent physical properties that can be used: volume of a liquid, length of a solid, pressure of a gas held at constant volume, volume of a gas held at constant pressure, electric resistance of a conductor, color of a very hot object, etc. Ex: The mercury thermometer is an example of a common thermometer that uses the variation of volume with temperature: The level of the mercury rises due to thermal expansion. Temperature can be defined by the height of the mercury column ice and water In order to quantify temperature, the thermometers are calibrated using different temperature degree scales such as Celsius C, Kelvin K or Fahrenheit F all of which are purely conventional boiling Ex: A thermometer using Celsius scale can be calibrated following the procedure: 1) Dip it into a mixture of ice and water at atmospheric pressure: the reading is assigned a value of 0 C. 2) Then, bring it to the temperature of boiling water: the reading is assigned a value of 100 C. 3) Impart the space between 0 and 100 into 100 segments, each representing a change of temperature of one C.

7 Temperature: The Kelvin Scale The Kelvin scale is the most common in science: it is defined in a fundamental way rather than based on the properties of a certain substance Ex: One way to define the Kelvin scale is by using a constant-volume gas thermometer: a flask of some gas kept at constant volume in thermal contact with the bath to be measured The bath temperature is varied and recorded (say in C), and the pressure in the gas is monitored using the height h of mercury column, P = P 0 + ρgh: the pressure will vary linearly with the temperature Irrespective of the starting pressure or the gas, if the lines are extrapolated to zero pressure, they will intersect in the same point which correspond to absolute zero temperature, C The Kelvin scale takes this temperature as zero, while one Kelvin is taken to be equal to one degree Celsius, such that the two scales are related by T = T C In SI, the unit Kelvin is defined in terms of absolute zero and the triple point of water the point at 0.01 ºC where water can exist as solid, liquid, and gas. Therefore Def: One Kelvin is 1/ of the temperature of the triple point of water

8 Temperature: Comparison between scales The most common scale used by public in the US is Fahrenheit scale related to the Celsius scale by 9 5 F 5 C C 9 F ( ) T = T + 32 T = T 32 Ex: Thus, we see that the temperatures of freezing water, of boiling water, and absolute zero are given respectively by 9 ( ) + = ( 100 C) + 32 = F ( ) C F Notice that, while one Kelvin and one Celsius degree are the same (since Celsius is just an offset of Kelvin scale), a change of temperature of one Fahrenheit degree is 9/5 times smaller than a change of one degree Celsius, that is the variations of temperature in the two scales are related by T = 5 T = 1.8 T 9 F C C Ex: If a quantity of water is warmed by 5 ºC, the change in temperature expressed in ºK is ( ) 9 T F = = 5 5 K 9 K C + 32= F

9 Thermal Expansion of Solids and Liquids: Linear and area expansion Recall that one way to build a thermometer is by using the temperature dependency of volume: objects expand or shrink with changing temperature The thermal expansion of an object is a consequence of the change in the average separation between its constituent atoms or molecules: as temperature increases, the amplitude of molecule vibration increases causing the object to expand Depending on the geometry of the system, the expansion can be in: 1. Linear: if an object of length L 0 at temperature T 0 experiences a small change in temperature T, it will change in length by α 0 L= L T 2. Superficial: if an object of area A 0 at temperature A 0 experiences a small change in temperature T, it will change in area by γ 0 A= A T 3. Volume: if an object of volume V 0 at temperature T 0 experiences a small change in temperature T, it will change in volume by β 0 V = V T The coefficients α, β and γ are called coefficients of linear, superficial and volumetric expansion, measured in 1/ C: they depend on temperature only very weakly, so they can be considered constant for small temperature variations Ex: a) Accounting for thermal expansion is crucial when building bridges, houses, railways, planes, etc. b) Global warming has an effect an increase in the volume of the oceans not only due to the polar ice melting, but also due to thermal expansion

10 Most liquids expand strongly with increasing temperature; however, between 0 C - 4 C, water decreases in volume with increasing temperature However, above 4 C the volume increases, such that ice floats (since it has a smaller density than water), and water-carrying pipes burst when they freeze Problems: The Unusual Behavior of Water Above 4 C, water increases in volume (decreasing density) like any other liquid 1. Alternative expressions for expansion relationships: Say that a long rod made of a material with coefficient of linear expansion α has a length L 0 at temperature T 0. The temperature is increased to T. What is the final length L of the rod in terms of α, L 0, T 0 and T? 2. Relationship between the expansion coefficients: Consider a block made of a material that thermally expands uniformly in all direction. What is the relationship between its linear, superficial and volumetric coefficients of thermal expansion? 3. Matching surfaces: A steel ring with a hole of area A 0s = 3.99 cm 2 is to be placed on an aluminum rod with cross-sectional area A 0a = 4.00 cm 2. Knowing the coefficients of superficial thermal expansion, γ s and γ a, what change of temperature of both will allow the ring to be slipped onto the end of the rod?

11 Ideal Gas: Characterization The ideal gas is a thermodynamic theoretical model used to emulate the behavior of most gaseous systems, and as a startup and reference for more complex models Ex: Most gasses at room temperature and atmospheric pressure behave like an ideal gas Macroscopic characteristics: 1. If not in a vessel, an ideal gas does not have a fixed volume or pressure 2. In a vessel, the gas expands to fill the container independent on the presence of another gas in the same container Microscopic characteristics: 1. Collection of atoms or molecules that move randomly 2. Each atom or molecule is considered point-like 3. The particles exert no long-range force on one another If the particles of an ideal gas contain only one atom, the gas is called monoatomic. If there are more bounded atoms per particle (molecules), the gas is called poliatomic Ex: Noble gasses, such as He, Ne, Ar, etc. are monatomic; H 2 or O 2 gasses are diatomic The ideal gas model can be extended to describe non-chemical systems such as the free electrons in metals, but the model fails to describe gasses at very low temperatures or high pressure, or heavy gasses such as water vapors

12 Ideal Gas: Measuring molar quantities It s convenient to express the amount of gas in a given volume in terms of the number of moles, n One mole of any substance is characterized by: 1. the same number of particles, called Avogadro number, A = particles/mol 2. a molar mass, M: for the monoatomic gasses M is given numerically by the atomic mass of the element expresses in gram/mol 3. a molar volume, V M : if the gas density is ρ, we have V M = M/ρ Consequently, the number of moles in a mass m of gas containing particles within a volume V is given by Exercise: molar calculations A container of volume 10 4 m 3 is filled with 20 grams of argon gas. a) How many moles of argon are in the container? m V n= = = M V A M b) How many argon atoms are in the container? c) What is the molar volume of argon?

13 Ideal Gas: Equation of state Imagine an experiment on n moles of ideal gas, allowing the independent modification of volume V, pressure P and temperature T Experimental observations: 1. isothermal (T = const.) PV = const. (Boyle s law) 2. isobaric (P = const.) V/T = const. (Charles s law) 3. isochoric (V = const.) P/T = const. (Gay-Lussac s law) These results can be integrated into the equation of thermodynamic state for the ideal gas given by n, V, P, T Ideal Gas Law: If n moles of ideal gas are confined in a volume V under a pressure P at a temperature T, then PV = nrt where R is the universal gas constant R = 8.31 J/mol.K, and T must be in Kelvins Alternative forms: m V PV = RT = RT = RT = kbt where k B is Boltzmann constant M V M A k B = R/ A = J/K

14 Problems: 4. Molar volume: What is the volume occupied by one mole of any ideal gas in the so called normal conditions, that is at atmospheric pressure P Pa and temperature 0 C? 5. Isochoric process: A spray can of volume V 0 =125 cm 3 containing a propellant gas at twice the atmospheric pressure, is initially at T 0 = 22 C. The can is tossed in an open fire. Assuming no change in volume, when the temperature in the gas reaches T = 195 C, what is the pressure P inside the can? 6. Thermal and mechanical equilibrium: A vertical cylinder of crosssectional area A = m 2 is fitted with a tight-fitting, frictionless piston of mass m = 5.0 kg. If there are n = 3.0 mol of an ideal gas in the cylinder at temperature T = 500 K, what is the height h at which the piston will be in equilibrium under its own weight?

15 Ideal Gas: Kinetic Theory The macroscopic properties of the ideal gas can be obtained using a model for the behavior of the gas at microscopic level, as predicated by the kinetic theory Assumptions made by the kinetic theory: 1.The number of molecules in the gas is large and the average separation between them is large compared to their dimensions (particles) 2. The molecules obey Newton s laws of motion, but as a they move randomly 3. The molecules interact only by short-range forces during elastic collisions 4. The molecules make elastic collisions with the walls 5. The gas under consideration is a pure substance: all the molecules are identical Then, it can be shown that the pressure P in the gas is proportional to the number of molecules per unit volume V and to the average translational kinetic energy of a molecule: Therefore, pressure can be increased by P V 2 1. Increasing the number of molecules per unit volume in the container 2 = mv 2. Increasing the average translational kinetic energy of the molecules, which can done by increasing the temperature of the gas How does temperature increase the average kinetic energy of the molecules?

16 Ideal Gas: Molecular interpretation of temperature Comparing the kinetic expression for the pressure with the equation of state for the ideal gas, we see that: 1. Temperature is proportional to the average kinetic energy per molecule PV = ( ) 3 mv mv = kbt 2 kbt = 2 mv PV = kbt 2. The total translational kinetic energy is proportional to the absolute temperature ( ) KEtotal = 2 mv = 2 kb T = 2 A kb T = 2 nrt 1 Therefore, since in a monoatomic gas the molecules cannot rotate, the net translational kinetic energy provides all the internal energy U of the ideal gas: Hence, we see that the internal energy of an certain amount of ideal gas depends only on temperature, that is, if the temperature is constant, the internal energy is conserved, and vice-versa However, be cautious: this formula works only for ideal gasses A = R U = 3 2 nrt

17 Ideal Gas: Molecular speeds By the fact that the internal energy of the ideal gas is given by the net kinetic energy, we can estimate the average speed of the molecules of individual mass m, expressed as the root-mean-square speed v rms : ( ) k B T U = 2 mv = 2 kbt v = m 2 3kBT 3RT = v = = m M Based on this result, we conclude that, at a given temperature, lighter molecules move faster, on average, than heavier ones The ideal gas molecule speeds are distributed as described by the Maxwell-Boltzmann distribution This bell-shaped curve peaks at the most probable speed v mp, and the peak flattens, broadens and shifts to the right as the temperature increases The area under the curve is actually the total number of molecules, so it doesn t change with temperature Notice that the average speed is not the most probable speed: mp rms v rms v > v > v

18 Problems: 7. Kinetic theory and internal energy: A container of constant volume V 0 = 2.5 liter confines n = 0.20 mol of helium gas at temperature T 0 = 10 C. The container is warmed up to temperature T = 60 C. a) How come even the molecules that do not make contact with the container wall in average increase their speed? b) Give examples of mechanical quantities that increase such that the pressure on the walls increases? c) Calculate the increase in internal energy of the gas d) Calculate the rms speed at T 0 T 0 T 8. Conservation of internal energy: A thermally isolated container of volume V and temperature T is separated by a piston into two equal compartments containing the same type of monoatomic ideal gas. However, the left half has n moles, and the right half has 2n moles. The piston is removed. a) What happens with the temperature of the gas and why? b) Write out the internal energy of the gas after expansion in terms of n, R and T c) Find the pressure in the expanded gas in terms of n, R, V and T T T n, P 1, V/2 2n, P 1, V/2 P, V