Medical Physics Lecture 11 PDF

Medical Physics Lecture 11 Nikolay Gulitskiy  MOLECULAR KINETIC THEORY. THERMODYNAMICS In all bodies there is a constant movement of particles. Characteristic feature of this movement is its disorder. This movement is called thermal....

Medical Physics Lecture 11 Nikolay Gulitskiy  MOLECULAR KINETIC THEORY. THERMODYNAMICS In all bodies there is a constant movement of particles. Characteristic feature of this movement is its disorder. This movement is called thermal. The particles of the body form a complex system It is impossible to describe the motion of each particle. Example: One cm3 of gas contains 1019 molecules. To describe their movement 3∙1019 equations are needed, if you write 1 per second (60 ∙ 60 ∙ 24 ∙ 365=3 ∙ 107), it will take 1012 years.  2 approaches to the description of complex systems Thermodynamic approach Statistical approach = Molecular Thermodynamics is a branch of physics that describes Kinetic Theory (MKT) processes in a system consisting of microparticles at the MKT is a branch of physics that macroscopic level. determines the averaged parameters of Macroscopic parameters: microparticles (velocity, energy) and ✓ Temperature Т establishes their relationship with ✓ Pressure p macroparameters. ✓ Volume V Laws of thermodynamics = principles of thermodynamics The basic concepts of thermodynamics are introduced on the basis of experiment, based on general laws, such as the law of conservation of energy. MKT operates with statistical methods based on theory of probability.  The basic equation of MKT allows us to calculate the pressure of an ideal gas on the vessel walls. Ideal gas is a model of a substance: molecules = point bodies, there are no forces of intermolecular interaction. Interaction between molecules only in collisions. Collisions are absolutely elastic. The pressure on the vessel wall arises due to the transmission of a momentum to it by gas molecules of mass m0 during collisions The change in the projection of the momentum of one molecule on the х axis is equal to: 𝑝2𝑥 − 𝑝1𝑥 = −𝑚0 𝑣𝑥 − 𝑚0 𝑣𝑥 = −2m0 vx By virtue of the law of conservation of momentum, the wall will receive an impulse: : ∆𝑝𝑥 = + 2m0 vx Averaged: : ∆𝑝𝑥 = + 2m0 𝑣𝑥  Derivation of the basic MKT equation One particle acts onaverage is the the wall with speedforce is the The average number of impacts on one wall speed the is average speed (half of the particles) volume v12, v22, v32,…in (*) is the average is the average speed concentration total pressure is the of allspeed average particles: speed 1 2 All directions of velocity in 𝑝 = 𝑛𝑚0 𝑣 the gas are equally probable. 3 the basic equation of MKT, 𝑣 is the average speed  Thermodynamic concept of temperature. Thermodynamic equilibrium is a state of an cold is the ambient average speed hot isolated (there is no exchange of energy and matter between the system and the environment) system, in which all its parameters have assumed the same values for all points of the system and do not change over time. Temperature is a parameter that is the same for all elements of a system in the state of thermodynamic equilibrium  Zeroth law of thermodynamics An isolated thermodynamic system in a nonequilibrium state eventually comes to the equilibrium when the temperatures of all macroscopic parts are the same. https://www.sites.google.com/site/opatpofizike/_/rsrc/1391261249193/uravnenie-mendeleeva-klapejrona/dalton.gif EXAMPLES: ✓ Mixing hot and cold liquids – the same temperature is set throughout the volume ✓ Mixture of gases – the partial pressure is the same throughout the volume, but it takes its own value for each gas. Only the temperature is the same for all gases of the mixture. Partial pressure is the pressure that a gas would produce if it alone occupied the entire volume equal to the volume of a mixture of gases.  Molecular kinetic interpretation of temperature It has been experimentally established that for different gases (characterized by different values of p, V, N) in the state of thermal equilibrium, the relation is fulfilled: 𝑝𝑉 = 𝑐𝑜𝑛𝑠𝑡 at a given temperature 𝑁 Then it makes sense to associate this expression with temperature: 𝑝𝑉 = 𝑘𝑇 where T is the absolute temperature, 𝑇 ≥ 0, 𝑁 𝑝𝑉 Because cannot be a negative value. 𝑁 (Т=0 when р=0 at a fixed volume or V→0 at p = const) T = t + 273, t – temperature on the Celsius scale. k = 1,38∙10-23 J/K – Boltzmann constant  Molecular kinetic interpretation of temperature 2 1 2 2 𝑚0 𝑣 2 The basic equation of MKT: 𝒑 = 𝑛𝑚0 𝑣 = 𝑛 = 𝒏𝜺0 3 3 2 3 Where 𝜀0 − average kinetic energy of one molecule 2 average kinetic energy of one molecule 𝑝 = 𝑛𝜀0 3 3 𝑝 3 𝑝𝑉 𝑁 𝜀0 = = because concentration 𝑛 = 2𝑛 2 𝑁 𝑉 𝑝𝑉 3 The temperature on the Kelvin scale is proportional = 𝑘𝑇 𝜀0 = 𝑘𝑇 to the average kinetic energy of one molecule. 𝑁 2 Temperature is a measure of the kinetic energy of molecules The proportionality coefficient between T and 𝜀0 allows you to measure the temperature not in J, but in more convenient units.  Boltzmann constant J Θ0 𝑇0 = = 273𝐾 𝑘 The Boltzmann constant is a coefficient that was introduced by Max Planck and named after the Austrian physicist Ludwig Boltzmann, one of the founders of statistical mechanics.  Two definitions of temperature Thermodynamic and statistical definitions of temperature do not contradict each other: In a state of thermodynamic equilibrium, the average kinetic energy of all molecules is the same. Otherwise, the molecules would exchange energy, which would contradict the requirement that all parameters remain unchanged over time.  The equation of state of ideal gas. The state of a given mass of an ideal gas is determined by the values of three macroparameters - pressure, volume, temperature. The equation connecting these three parameters is called the equation of state. 2 basic equation of MKT 𝑝 = 𝑛𝜀0 3 3 what is consistent with the thermodynamic 𝑝𝑉 𝜀0 = 𝑘𝑇 p = nkТ experimental law 𝑁 = 𝑘𝑇 𝑁2 𝑛 = − concentration of particles: => pV = NkT 𝑉 N= νNA, ν – number of moles, NA = 6,02∙10-23 mole-1. Then: 𝑚 pV = νNA kT 𝑝𝑉 = 𝑅𝑇 Equation of state of mass m of an ideal gas µ (Mendeleev-Clapeyron) where m/μ = ν, μ – molar mass NA k =R, R = 8,31 J/(mol*K) is the universal gas constant  Notes NOTE 1: p = nkT => pV = NkT=> Avogadro's law: equal volumes of gases at equal temperatures and pressures contain the same number of molecules. NOTE 2: The equation of state was originally obtained empirically (thermodynamically) ✓ Clapeyron (1834) for a given mass of a given gas: 𝑝𝑉 = 𝑐𝑜𝑛𝑠𝑡 (∗) 𝑇 ✓ by Mendeleev (1874) for one mole of any gas (by substituting the molar volume Vm instead of V in (*)): pVm = RT (**) Combining (*) and (**) gives the Mendeleev-Clapeyron equation of state. NOTE 3: The Mendeleev-Clapeyron equation is universal in nature, it does not include any quantity that characterizes the nature of the gas  Thermodynamic processes Any change in the macroscopic parameters in the transition of a system from one equilibrium state to another, i.e. from one set of parameters 𝑇1,𝑃1,𝑉1 to the other 𝑇2,𝑃2,𝑉2 is called the process (thermodynamic) or by transition from state 1 to state 2. process (transition) from state 1 to state 2 process  Isoprocesses Isoprocesses are processes in which one of the parameters - pressure, volume or temperature - remains constant, and only the other two change. The laws describing isoprocesses were discovered experimentally long before the derivation of the equation of state of an ideal gas. However, they can be derived from this law theoretically Gay-Lussac Charles Boyle–Marriott Law Law Law  Boyle–Marriott Law Isothermal process - Boyle–Marriott Law. For a given mass of gas, the product of the gas pressure by its volume is constant if the gas temperature does not change. T = const, pV = const ⇓ p(V) – hyperbole Isotherms in coordinates pV, pT, VT Different isotherms correspond to different temperatures. In order for the temperature to remain unchanged, it is necessary that the gas can exchange heat with an external system (thermostat).  Isobaric process – Gay-Lussac Law For a gas of a given mass, the ratio of the volume of the gas to its temperature remains constant if the gas pressure does not change p = const 𝑉 = 𝑐𝑜𝑛𝑠𝑡 𝑇 ⇓ V(T) is a straight line Isobars in coordinates pV, pT, VT Different pressures correspond to different isobars. The Gay-Lussac law is not observed in the region of low temperatures close to the temperature of liquefaction (condensation) of gases.  Isochoric process - Charles' law: For a given mass of gas, the ratio of gas pressure to its temperature remains constant if the volume of gas does not change. V = const 𝑝 = 𝑐𝑜𝑛𝑠𝑡 𝑇 ⇓ p(T) is a straight line Isochores in coordinates pV, pT, VT Different volumes correspond to different isochores. Charles' law is not observed in the region of low temperatures, close to the temperature of liquefaction (condensation) of gases.  Internal energy U The total energy of the system consists of mechanical and internal energy. In thermodynamics, bodies at rest are usually considered, so the mechanical energy E does not change. The internal energy of a substance U consists of a combination of the kinetic energy of the chaotic motion of the particles that make up the substance and the potential energy of their interaction. For the ideal gas, the internal energy is determined only by the total kinetic energy of its molecules. ⇓ The internal energy of an ideal gas is characterized by temperature.  Ways to change the internal energy of the system Performing external force work on the system - macroscopic method (friction, deformation, compression of gas in the cylinder using a piston); In the process of heat exchange – microscopic method (interaction of particles, microprocesses) The amount of heat Q. The measure of the change in internal energy during heat exchange is the amount of heat. Types of heat transfer: Thermal Conductivity Radiation Convection  The first law of thermodynamics In accordance with the two ways of changing the internal energy, we can write: ∆U = Q + Aext, where Aext is the work of external forces performed on the gas (for example, the gas is compressed by a piston). If the gas does the work itself: ∆U = Q – A where A is the work performed by the gas itself against external forces (for example, the gas expands and moves the piston against the friction force). Formulation of the first principle of thermodynamics: The heat transmitted to the system is spent on changing its internal energy and on performing work by the system: Q = ∆U +A The impossibility of creating a perpetual motion machine follows from the first law.  Internal energy is a function of state of the system This means that each state of the system uniquely corresponds to a certain internal energy. It can be shown that the amount of internal energy depends only on temperature and does not depend on how the system came to this state. It follows from this that with any method of transition of the system from the first state to the second, its internal energy changes ∆U =U2 - U1 it will be the same and equal to the difference in the values of the internal energies of the system in these states. Quantities that do not depend on the background of the system and are determined by its state at the moment are called functions of state: U, T, P, V, etc.  The work on changing the volume of gas = the area under the curve p(V) Elementary work performed by gas when moving the piston at a distance dl Elementary work The total work performed by the gas when its volume changes from V1 to V2 is found by integrating;  AI ≠ AII → the work is not a function of state SII = AII of the system The work and the heat are the functions of processes, but not of the state. SI = A I IMPORTANT: A > 0 if it is performed by the system (volume increases), and A< 0 if the work is performed on the system by external forces (compression, volume decreases). Heat (dQ) is positive if the body receives heat, and negative if it gives.  Adiabatic process The processes occurring in a system that does not exchange heat with surrounding bodies are called adiabatic. The first law for the adiabatic process (dQ=0): Poisson equation dA = - dU pVγ = const 𝐶𝑝 𝑑𝑄 where 𝛾 = , c= - heat capacity 𝐶𝑉 𝑑𝑡 The heat capacity C, as well as the heat Q, is a function adiabate 𝑪 > 𝑪 ⟹ 𝜸 > 𝟏: of the process (since Q includes A) 𝒑 𝑽 at V = const: А=0 and from the 1st law of dU thermodynamics follows: dQV = dU CV = ( )V isotherm dT At Dividing by dT, we get 𝜸 > 𝟏 => The adiabate of an ideal gas, constructed in ordinates p and V, always goes steeper than the isotherm  Work and heat in various processes 1. Isochoric process : V = const 3. Isobaric process : p = const 𝑉2 dV = 0 → A = 0 → dQ = dU 𝐴 = න 𝑝𝑑𝑉 𝑉1 2. Isothermal process : Т = const p = const, → A = p(V2-V1) dU = 0 → dQ = dA = pdV 𝑉2 p(V2-V1) = R(T2-T1) ⇓ 𝐴 = න 𝑝𝑑𝑉 𝑉1 R is numerically equal to the work performed by one mole 𝑅𝑇 of gas when heated by one degree in an isobaric process. Note that from the Mendeleev-Klayperon equation 𝑝 = 𝜈 , 𝑉 then: 4. Adiabatic process : dQ = 0 𝑉2 𝑅𝑇 𝑉2 𝐴= නν 𝑑𝑉 = 𝑣𝑅𝑇 ∙ 𝑙𝑛 dA = - dU = -vcVdT , cV - molar heat capacity 𝑉 𝑉1 𝑉1 𝑇2 𝐴 = − න 𝑣𝑐𝑉 𝑑𝑇 = −𝑣𝑐𝑉 (𝑇2 − 𝑇1 ) 𝑇1  THE SECOND LAW OF THERMODYNAMICS. ENTROPY Many phenomena are possible in principle, because they do not violate conservation laws, in practice they are not observed (very unlikely). TYPICAL EXAMPLES: Mixing of tea with water, the reverse process is difficult to imagine dissolution of ink in water and the inability to assemble again into a drop establishment of equilibrium temperature during thermal contact of hot and cold bodies, mixing of initially separated gases (diffusion) expansion of gas into the void a slowly deflating balloon: there is no chance that the air from the room will return back to the balloon The second law of thermodynamics is associated with the concept of entropy, a quantity that characterizes the direction of various processes  Macroscopic (thermodynamic) interpretation of entropy The first concept of entropy S was proposed by the German physicist Rudolf Clausius in 1862. He introduced this value in such a way that it can be used to calculate the function of process— the amount of heat, just as another process function - work is expressed in terms of macroparameters p and V. Clausius determined the entropy differential (its small change) dS as the reduced heat: where dQ is the elementary amount of heat received by the body, dQ/T - reduced heat, T is the temperature of source of the supplied heat : the process is carried out isothermically (T = const)  𝑑𝑄 Entropy 𝑑𝑆 = 𝑇 This definition is valid only for reversible processes. Then: dQ= TdS (by analogy with dA = pdV)  Reversible processes The process 1-> 2 is reversible if it is possible to perform the reverse process 2 —> 1 through the same intermediate states so that no changes occur in the surrounding bodies. Processes are called irreversible when a spontaneous reverse transition from the final state to the initial one through the same intermediate states as in the direct process is impossible. Nonequilibrium processes are always irreversible. EXAMPLE: gas under the piston. If you lift the piston very quickly, then at the first moment all the molecules will remain in the lower part of the vessel — nonequilibrium process. In order for such a process to go in the opposite direction, all the molecules must first gather in the lower part of the vessel, and then the piston must fall on them. Reversible process = equilibrium without friction. EXAMPLE: mechanical vibrations of motion without friction. A reversible process is a physical abstraction. All real processes are irreversible, at least because of the presence of friction force, which causes heating of the surrounding bodies. Reversible processes are an idealization that is convenient for solving many important issues and is a good approximation for practical calculations.  Properties of Entropy 𝑑𝑄 𝑑𝑆 = 𝑇 An important concept of entropy gain follows from Clausius' definition: Entropy is a function of state the process T1->T2 is divided into elementary subprocesses with dQ and Т=сопst EXPLANATION: 1st law of thermodynamics can be rewritten: TdS = dU + dA, then for one mole: since UIG =νcVT and dU= νcVdT , while dA=PdV. Entropy is an additive quantity: The entropy of a system is the sum of the entropies of all its parts  Microscopic interpretation of entropy (1877) I. Boltzmann gave a statistical (probabilistic) definition of entropy: Entropy is proportional to the logarithm of the probability N (statistical weight) of finding a system in a particular state. EXAMPLE: playing dice S = k lnN Macro state: Σ=3A set that implements this macrostate is unique: N=1. The system is fully defined, the entropy is zero S=0 There is no uncertainty in such a system. Macro—state: Σ=4d Sets implementing this macro-state: N=3. Entropy increases: S = kln 3 Uncertainty has appeared in the system The larger the value of the sum, the more different ways and hence more uncertainty. The state of the system it is becoming increasingly difficult to guess which numbers fell out on the dice.  Closed and open systems An open system is one that can exchange matter or energy with the environment. Example: an open glass – water evaporates. A closed system is a system that cannot exchange matter or energy with the environment. Example: thermos - water does not evaporate and does not cool down. The second principle of thermodynamics is formulated for closed systems. (Classical thermodynamics considers closed systems)  Thermodynamic formulations of the Second law of thermodynamics Heater The second law establishes the qualitative disparity of different types of energy in the sense of the ability to transform into other types. He has several formulations. ❖ Rudolf Clausius (1850): The process of spontaneous transfer of heat from a Cooler less heated body to a more heated one is impossible (without the influence of other bodies!)  W. Thomson Heater William Thomson (for scientific merit — Lord Kelvin (1851): A cyclic process is impossible, the only result of which would be the absorption of heat from the heater and its complete transformation into work=> A perpetual motion machine of the second kind is impossible. explanation: The device in which the working fluid (gas) receives heat from the heater, spends part of it on performing work (expanding), Heater and gives the other part to the cooler (to shrink and return to its original state) is called a heat machine. Heat transfer to the cooler is a prerequisite. EXAMPLE: any organism can be considered as a heat engine. working HEATER -nutrients (energy Q1 is released when they are split). body Q2 — heat released into the environment, which is a COOLER. Cooler  The second law of thermodynamics and entropy: dS > O— entropy in a closed system The second law of thermodynamics states that if a system is closed, then entropy in it cannot decrease. I.e., processes occur spontaneously in complex systems that do not lead to a decrease in entropy. In the absence of external influence, a complex system is disordered, i.e. it tends to thermodynamic equilibrium (without homogeneities), or remains unchanged. Shaking the balls Chaotically scattered balls is more probable Before shaking after shaking their condition: it can be obtained after shaking, there are more ways than when they are stacked in an orderly manner. NOTE: shaking is analogous to the thermal motion of molecules