Thermodynamics: Microscopic vs. Macroscopic (Chapters 16, )

Transcription

1 Thermodynamics: Microscopic vs. Macroscopic (Chapters 16, ) Matter and Thermal Physics Thermodynamic quantities: Volume V and amount of substance Pressure P Temperature T: Ideal gas Zeroth Law of Thermodynamics Temperature scales Phase diagrams Macroscopic description: Thermodynamic processes PV diagrams Micro/Macro description: the kinetic theory of gases Molecular speeds Equipartition Theorem

2 Matter Phases Physics deals with the states, processes and interactions involving matter in various phases (solid, liquid, gas, plasma). The study of the various phases and scales of matter splits Physics into various branches with specific methods Phase transitions driven by energy transfer determine a restructuring of the net microscopic energy content of the substance, called: Def: The internal energy E of an object is the net energy necessary to build it from its elementary constituents: the kinetic energy K of its micro-parts and the potential energy U associated with the various bonds Sublimation E K U Solid Melting Liquid Boiling Gas Freezing E includes the atomic vibrational and rotational kinetic energy, and the electric or nuclear potential energy of atoms, molecules and lattice E includes the atomic translational, vibrational and rotational kinetic energy, and the potential energy of atoms and molecules Condensation Added energy (heating) E includes the atomic translational kinetic energy (plus vibrational and rotational for a molecular gas), and the potential energy of atoms

3 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 matter Ex: Say that the moving box discussed on one of the previous slides is metallic: the heat released by 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 describes the various phases based on the interdependence of the macroscopic and microscopic scales of matter For instance, it uses statistical mechanics based on probability theory applied to large ensembles of particles to model macroscopic properties inferred from averaged microscopic mechanical, electromagnetic and quantum mechanical characteristics of the atomic and molecular constituents of the system Besides internal energy, the macroscopic thermodynamic state of a simple system is given by a relationship between several thermodynamic parameters: 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

4 Thermodynamic parameters Volume and Quantity of substance Solids and liquids occupy fixed volumes of space, while gases take the volume of the containing vessel The mass density describes how a substance of mass m is distributed within the respective shape of the object Thus, ff the mass is distributed uniformly within a volume V, or plane of area A, or along a line of length L, we define volumetric mass density: superficial mass density: linear mass density: mv ma ml In the case of gases, it is customary 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, N A = particles/mol 2. a molar mass, M: for the monoatomic gases 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 N particles within a volume V is given by m N V n M N V A M

5 Quiz: 1. Which contains more molecules, a mole of hydrogen gas (H 2 ) or a mole of water vapors H 2 O? a) The hydrogen. b) The water. c) They each contain the same number of molecules. 2. Consider two diatomic gases, hydrogen H 2 and oxygen O 2, containing equal number of molecules. What is the relationship between the masses of the two gases? a) m H = m O /8 b) m H = 8m O c) m H = m O /16 d) m H = m O Exercise 1: molar calculations A container of volume V = 10 4 m 3 is filled with a mass m = 20 grams of argon gas. a) How many moles of argon are in the container? b) How many argon atoms are in the container? c) What is the molar volume of argon?

6 Thermodynamic parameters Pressure Def: The pressure P associated with a force F exerted uniformly across a surface A is F 2 P N m Pascal Pa A Hence, the pressure determines how the effects of applying a force will be different depending on how it is distributed on the surface Ex: Pressure in gases gases contain a large number of molecules moving randomly. When in a vessel, the gas fills the entire available volume The pressure is due to the collisions of molecules with the walls of its container: as each molecule strikes the wall, its momentum changes due to the force F w m acted by the wall paired with a force F m w acted by the molecule on the wall: Newton s 3 rd Newton s 2 nd p pafter pbefore Fm w Fw m t t Fm w The pressure in a thermally stable gas is the same in every point throughout the gas Dalton Law: The net pressure in a gas mixture is equal to the sum of the partial pressures exerted by each gas independently occupying the entire volume of the container p after Fw m p before

7 Exercise 2: 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?

8 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

9 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.

10 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

11 Temperature Comparison between scales The most common scale used by public in the US is Fahrenheit scale related to the Celsius scale by T T 32 T T F 5 C C 9 F Ex: Thus, we see that the temperatures of freezing water, of boiling water, and absolute zero are given respectively by C F 0 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 1.8T 9 F C C Ex: If a quantity of water is warmed by 5 ºC, the change in temperature expressed in ºF is 9 T F 5 5 F 9 F C F

12 Quiz: 3. Three gases are cooled down at constant volume, while the pressure in the gas is monitored. The pressure winds out varying with temperature as on the figure. Which of the following is true about the temperature dependency of the pressures P i =1,2,3 in the three gases? a) dp1 dp2 dp3 273 dt dt dt b) P at 273 (where a i are volume and gas dependent constants) i i c) P i i a T An object is warmed up. Which of the following is true about the relationship between the change in temperature expressed in Celsius, Fahrenheit and Kelvin scales? a) b) c) T T T 9 C F 5 K T T T 5 C K 9 F T T T C F K

13 Temperature Phase transitions Pressure and temperature allow a more nuanced description of phase transitions since each transition will occur at critical temperatures depending on the pressure Hence, a substance can be characterized using a phase diagram: a P vs. T map showing regions of uniform phase bordered by boundaries of critical temperatures at various pressures Ex: Phase diagram of water Notice that the familiar critical temperatures of water (0 C for melt/freeze and 100 C for boil/condense) are true only at a normal atmospheric pressure P 0 = 1 atm At lower pressures (such as when climbing at higher altitudes) the interval between freeze and boil shrinks For lower pressures (P < atm), the water cannot exist in liquid form at any temperature: ice sublimates, that is, changes into vapors directly Notice the point (0.006 atm, 0.01 C) called the triple point of water where all three phases coexist Quiz: 5. What happens if, at a certain temperature, the pressure of liquid water is suddenly dropped from above 1 atm? a) The water definitely freezes b) The water definitely boils c) Depending on temperature, the water may either freeze or boil

14 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 low density gases well above their condensation points 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 polyatomic Ex: Noble gases, such as He, Ne, Ar, etc. are monatomic; H 2 or O 2 gases 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 gases at very low temperatures or high pressure, or heavy gases such as water vapors

15 Ideal Gas Ideal gas processes 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 N PV RT RT PV NkBT where k B is Boltzmann constant M N A k B = R/N A = J/K

16 Ideal Gas PV diagrams It is customary to represent ideal gas processes on P vs. V graphs called PV diagrams Processes of n moles of gas are paths between states characterized by sets of parameters (P, V, T) The processes are considered slow or quasi-static, such that each point on a path is in thermal equilibrium Temperatures can be shown using isotherms: sets of hyperbolas characterized by constant temperature There are infinitely many paths between any two states, and they are reversible: that is, they can be traveled in both directions Ex: PV diagrams for different types of processes Isobaric: Isochoric: Isothermal: P = const. V = const. T = const. P compression P P V i =V f T i =T f expansion P i =P f V V V V Vi f Pf Pi PV i i Pf Vf Ti Tf T T i f P P 1 P 2 Isotherms Path V 1 Adiabatic: the graph is steeper than an isotherm P PV P V i i f f P 1,V 1,T 1 V V 2 P 2,V 2,T 2 P T 4 T 3 T 1 T 2 Increasing temperature Generic: PV i T i i V V PV f T f f

17 Quiz: 6. A cylinder of gas has a frictionless but tightly sealed piston of mass M. A small flame heats the cylinder, causing the piston to slowly move upward. For the gas inside the cylinder, what kind of process is this? a) Isobaric. b) Isochoric. c) Isothermal. 7. A cylinder of gas floats in a large tank of water. It has a frictionless but tightly sealed piston of mass M. Small masses are slowly placed onto the top of the piston, causing it to slowly move downward. For the gas inside the cylinder, what kind of process is this? a) Isobaric. b) Isochoric. c) Isothermal. Exercise 3: PV diagrams The blue curves on the adjacent diagram are isothermals of a monoatomic ideal gas, with equal temperature differences between them. Identify each of the thermodynamic paths represented on the figure as isobaric, isothermal, isochoric or adiabatic. P B A C D V

18 Problems: 1. 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? 2. An 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? 3. Thermal and mechanical equilibrium: A vertical cylinder of cross-sectional area A is fitted with a tight-fitting, frictionless piston of mass m. There are n mols of an ideal gas in the cylinder at temperature T 0. a) What is the height h 0 at which the piston is in mechanical equilibrium? b) Say that the piston is slowly pushed in by a constant force to a gas height H. What is the final temperature T of the gas? c) Say that the cylinder is dipped in a water tank at temperature T 0. Then the piston is pushed in just a bit and then released. Are the ensuing oscillations simple harmonic? 4. Chain of thermodynamic processes: A quantity of n = 2 mol of monoatomic ideal gas first expands isothermally from V 0 = m 3 to V 1 = m 3 at a temperature of T 0 = 0 C, and then it is returned to the original volume V 0 by means of an isobaric process. Then the system returns to the original state via the shortest path on the PV-diagram. a) Sketch the PV diagrams of all processes b) Calculate the unknown parameters for each of the visited states.

19 Kinetic Theory Assumptions 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 about the molecules in a gas: 1.Their number is large and the average separation between them is large compared to their dimensions 2. They obey Newton s laws of motion, and move randomly 3. They interact only by short-range forces during elastic collisions 4. They interact with the walls by elastic collisions 5. The gas under consideration is a pure substance: all the molecules are identical So, individual molecules travel in zigzag: the average distance traveled between successive collisions is called mean free path λ, and depends on the concentration of particles (N/V) and the size of each molecule such as the radius r if the molecules are considered spherical: N V r

20 Kinetic Theory Molecular velocity The molecule speeds in an ideal gas are distributed as described by Maxwell-Boltzmann distribution: a bell-shaped curve that peaks at the most probable speed v mp The peak flattens, broadens and shifts to higher v mp as the gas temperature increases Notice that the area under the curve is actually the total number of molecules N in the sample so it doesn t change with temperature The motion of the molecules can be also described using the average speed v avg of the molecules or their root-mean square speed v rms given by the square of the average of the speed squares: v rms So, the average speed is not the most probable speed: v 2 v v v mp avg rms Ex: Speed distribution in a sample of nitrogen gas (N 10 5 atoms) Equal areas (N) 300 K Most probable speed v mp v avg v rms 900 K

21 Quiz: 8. A gas experiences an isobaric increase in temperature. What happens with the mean free path of the molecules? a) Increases. b) Decreases. c) Doesn t change N V r Exercise 4: Molecular speeds A six-molecule gas is confined in a container. A snapshot of the gas reports the velocity vectors shown in the figure. Calculate the average velocity, average speed and root-mean square speed.

22 Kinetic Theory Deriving macro parameters from micro analysis Using the simplifying assumptions about the motion of large ensembles of molecules in the gas, one can derive a relationship between the macro-characteristics (pressure P and volume V) and micro-characteristics of the gas (such as the average translational kinetic energy of a molecule): Physical situation: consider a volume V of ideal gas with N spherical molecules, each of mass m. As we learned earlier, the pressure in the gas is due to collisions. Consider N c such molecules within a distance Δx from a wall of area A. Each time Δt, the average force on the wall is p px _ after px _ before Nc F Nc Nc 2mvx t t t In the time Δt between two successive collisions, each molecule travels distance 2Δx. Hence, N c /Δt comes from N N N N N Av Ax Av t V V c c c 1 x 2 t 2 x z F N V p y p x _ before mv x _ after x x v t 1 x 2 2 Amv x Therefore, since the molecules move identically in all directions, the pressure on the wall is F N 2 2 N 1 2 P mv P 3 V 2 mvrms x 2 N A V P 2 3 V avg v 2 mvr ms avg x vy vz 3v 3v rms mv x A x F Average translational kinetic energy per molecule

23 Kinetic Theory 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 translational kinetic energy per molecule 2 Kinetic Theory PV N avg N Nk T T 3k 3 B kb 2 mv Ideal Gas Law PV NkBT 3 2 avg B avg rms 2. The net kinetic energy of N molecules is proportional to the absolute temperature 3 3 N 3 Knet N avg 2 NkBT 2 N AkB T 2 nrt N A Therefore, in a monoatomic ideal gas where molecules cannot rotate or vibrate, and there are no potential energies the net translational kinetic energy provides all the internal energy E of the ideal gas: R E nrt Nk T B Hence, we see that the internal energy or thermal 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: these particular formulas work only for ideal gases

24 Thermal Energy In Gases and Solids Notice that the thermal energy per particle of monoatomic gas contains the same amount of energy attributed to each direction available for translation, x, y or z: avg 2mvrms 3 2kBT The available distinct and independent motions or configurations of an object are called degrees of freedom. So we see that a monoatomic gas gets ½k B T energy per each available translational degree of freedom. This predication can be generalized by: The Equipartition Theorem: At thermal equilibrium, the net internal energy stored in a system is evenly shared by all its available modes, or degrees of freedom Ex: Thermal energy equipartitioned in gases and solids Diatomic gas: Deg. of freedom: 3 transl. + 2 rot z y x For N molecules: z y x Simple Solid: Deg. of freedom: 3 vibr. Bonds: 3 Total: 6 modes For N atoms: E Navg 2NkBT 2nRT E Navg 2 NkBT 3 NkBT 3nRT

25 Kinetic Theory Molecular speeds and temperature By the fact that the internal energy of the ideal gas is given by the net kinetic energy, we can estimate the temperature dependency of the root-mean-square speed v rms : kT B 3kT B 3RT E N 2mvrms 2NkBT vrms vrms m m M Based on this result, we conclude that, at a given temperature, lighter molecules move faster, on average, than heavier ones Quiz: 9. A rigid container holds both hydrogen gas (H 2 ) and nitrogen gas (N 2 ) at 100 C. Which statement describes the average translational kinetic energies of the molecules? a) ε avg of H 2 < ε avg of N 2. b) ε avg of H 2 = ε avg of N 2. c) ε avg of H 2 > ε avg of N How about the rms speeds of the molecules in the gases above? a) v rms of H 2 < v rms of N 2. b) v rms of H 2 = v rms of N 2. c) v rms of H 2 > v rms of N 2.

26 Problems: 5. Kinetic theory and internal energy: A vessel of volume V 0 = 2.5 liter contains 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 increase their speed in average? 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 6. 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 final internal energy of the gas 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