DOCSLIB.ORG

Equation of State of Ideal Gases

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Equation of state of ideal gases Objective For a constant amount of gas (air) investigate the correlation of 1. Volume and pressure at constant temperature (Boyles law) 2. Volume and temperature at constant pressure (Gay-Lussac's law) 3. Pressure and temperature at constant volume Charles's law Introduction The gas laws are thermodynamic relationships that express the behavior of a quantity of gas in terms of the pressure P , volume V , and temperature T . The kinetic energy of gases expresses the behavior of a "perfect" or ideal gas to be PV = nRT , which is commonly referred to as the ideal gas law. However, the general relationships among P , V and T contained in this equation had been expressed earlier in the classical gas laws for real gases. These gas laws were Based on the empirical observations of early investigators, in particular the English scientist Robert Boyle and the French scientists Jacques Charles and Joseph Gay-Lussac. Theory I Boyle's law: Around 1660, Robert Boyle had established an empirical relationship between the pressure and the volume of a gas, which is known as Boyle's law: At constant temperature, the volume occupied by a given mass of gas is inversely proportional to its pressure. In mathematical notation, we write 1 V / ; (1) P In equation form, we may write k ¢V = : (2) P Where k is the constant of proportionality for a given temperature. The pressure of a con¯ned gas is commonly measured by means of a manometer. The gas pressure for such an open-tube manometer is the sum of the atmospheric pressure Pa and the pressure of the height di®erence of mercury P = Pa + ½m £ g £ hm: (3) 1 but Pa = 76 cm Hg = ½m £ g £ 76 then equation (3) becomes: P = ½m £ g £ (hm + 76): (4) 3 2 where ½m = 13:6 gm=cm and g = 980cm=s The volume of the enclosed gas is the volume of the measuring tube segment marked in brown added to the volume calculated from the length of the column of air: V = Vl + 1:01 ml (5) or V = ¼r2l + 1:01 ml µ ¶ 1:14 2 = ¼ l + 1:01 ml 2 = 1:02l + 1:01 »= l + 1 = l0 (6) Substituting with P and V in eq. (2): 0 PV = ½m £ g £ (hm + 76) £ l = k: (7) Let k k0 = ½m g then Boyle's law is k0 h = ¡ 76 (8) m l0 Where hm represents the change in pressure, and since the cross sec- tional area of the tube is constant,then l0 represents V . II Charle's and Gay-Lussac's laws : In 1787, the French physicist Jacques Charles reported the result of a series of experiments which is known as Charles' law : At constant volume, the pressure exerted by a given mass of gas is proportional to its absolute temperature. In mathematical notation, P / T; (9) or P = k T: (10) 2 where k is a constant of proportionality. If we substitute for the value of P from eq(4 ), we get ½m £ g £ (hm + 76) = k T: (11) or k hm + 76 = £ T: (12) ½m g then 0 hm = k T ¡ 76: (13) Notice that the gas laws are expressed in terms of absolute tempera- ture (Kelvin). Where TK = Tc + 273 In 1802, Gay-Lussac, using a somewhat di®erent experimental ap- proach, essentially restarted Charles' law in a form more commonly used today (also known as Gay-Lussac's law): At constant pressure, the volume of a given mass of gas is proportional to its absolute temperature. That is, V / T; (14) or V = k00 T: (15) Substituting for the value of V from eq.(6) l0 = k00 T: (16) Equipments Gas laws apparatus, lab thermometer (¡10::: + 100c), distilled water. Procedure ² PART (I) 1. During the experiment, the temperature in the measuring tube must be kept constant. Keep the heater o® and record reading of the thermometer in the air tube. 2. raise or lower the open end of the tube until the mercury levels in both sides are equal, then record the length of the enclosed air 0 column l , where the di®erence of mercury levels hm = 0. Repeat this step several times 3 3. Raise the mercury level in the open tube to decrease the air column, then record l; hm. 1 4. Plot hm on the y-axis, and l on the x-axis, then calculate the slope. ² PART(II) 1. Set the thermostat to 30 c± wait until the thermometer read 30 c±. 2. Raise or lower the open end of the tube until the mercury levels in both sides are equal, and then record the length of the enclosed 0 0 air column l = l0, and the di®erence of mercury levels hm = 0. 3. During the experiment the volume must be constant; this means l0 must be constant. Increase the thermostat by 5 c± wait for temperature constancy in the measuring tube and the increment of l0 4 4. Raise the open end tube to return the length of air column to 0 0 l = l0, then record hm and TK . 5. ) repeat steps 3, 4 several times. 6. Plot hm on the y-axis, and TK on the x-axis, calculate the slope. ² PART(III) 1. Set the thermostat to 30 c± wait until the thermometer read 30 c±. 2. Raise or lower the open end of the tube until the mercury levels in both sides are equal, and then record the length of the enclosed 0 0 air column l = l0, and the di®erence of mercury levels hm = 0. 3. During the experiment the pressure must be constant; this means ± hm = 0. Increase the thermostat by 5 c wait for temperature constancy in the measuring tube and the increment of l0. 4. Lower the open end tube to return to equal levels of mercury 0 hm = 0, then record l and TK . 5. Repeat steps 3, 4 several times 0 6. Plot l on the y-axis, and TK on the x-axis, calculate the slope. Results Data table (I ): T0 = :::::::: l(cm) l0(cm) hm(cm) Data table (II ): 0 l0 = ::::::::; l0 = :::::::: Tc TK hm(cm) 5 Data table (II ): T0 = :::::::: Tc TK l(cm) l0(cm) 6.
Recommended publications
  • Thermodynamics Notes

    Thermodynamics Notes

    Thermodynamics Notes Steven K. Krueger Department of Atmospheric Sciences, University of Utah August 2020 Contents 1 Introduction 1 1.1 What is thermodynamics? . .1 1.2 The atmosphere . .1 2 The Equation of State 1 2.1 State variables . .1 2.2 Charles' Law and absolute temperature . .2 2.3 Boyle's Law . .3 2.4 Equation of state of an ideal gas . .3 2.5 Mixtures of gases . .4 2.6 Ideal gas law: molecular viewpoint . .6 3 Conservation of Energy 8 3.1 Conservation of energy in mechanics . .8 3.2 Conservation of energy: A system of point masses . .8 3.3 Kinetic energy exchange in molecular collisions . .9 3.4 Working and Heating . .9 4 The Principles of Thermodynamics 11 4.1 Conservation of energy and the first law of thermodynamics . 11 4.1.1 Conservation of energy . 11 4.1.2 The first law of thermodynamics . 11 4.1.3 Work . 12 4.1.4 Energy transferred by heating . 13 4.2 Quantity of energy transferred by heating . 14 4.3 The first law of thermodynamics for an ideal gas . 15 4.4 Applications of the first law . 16 4.4.1 Isothermal process . 16 4.4.2 Isobaric process . 17 4.4.3 Isosteric process . 18 4.5 Adiabatic processes . 18 5 The Thermodynamics of Water Vapor and Moist Air 21 5.1 Thermal properties of water substance . 21 5.2 Equation of state of moist air . 21 5.3 Mixing ratio . 22 5.4 Moisture variables . 22 5.5 Changes of phase and latent heats .
  • Thermodynamics

    Thermodynamics

    ME346A Introduction to Statistical Mechanics { Wei Cai { Stanford University { Win 2011 Handout 6. Thermodynamics January 26, 2011 Contents 1 Laws of thermodynamics 2 1.1 The zeroth law . .3 1.2 The first law . .4 1.3 The second law . .5 1.3.1 Efficiency of Carnot engine . .5 1.3.2 Alternative statements of the second law . .7 1.4 The third law . .8 2 Mathematics of thermodynamics 9 2.1 Equation of state . .9 2.2 Gibbs-Duhem relation . 11 2.2.1 Homogeneous function . 11 2.2.2 Virial theorem / Euler theorem . 12 2.3 Maxwell relations . 13 2.4 Legendre transform . 15 2.5 Thermodynamic potentials . 16 3 Worked examples 21 3.1 Thermodynamic potentials and Maxwell's relation . 21 3.2 Properties of ideal gas . 24 3.3 Gas expansion . 28 4 Irreversible processes 32 4.1 Entropy and irreversibility . 32 4.2 Variational statement of second law . 32 1 In the 1st lecture, we will discuss the concepts of thermodynamics, namely its 4 laws. The most important concepts are the second law and the notion of Entropy. (reading assignment: Reif x 3.10, 3.11) In the 2nd lecture, We will discuss the mathematics of thermodynamics, i.e. the machinery to make quantitative predictions. We will deal with partial derivatives and Legendre transforms. (reading assignment: Reif x 4.1-4.7, 5.1-5.12) 1 Laws of thermodynamics Thermodynamics is a branch of science connected with the nature of heat and its conver- sion to mechanical, electrical and chemical energy. (The Webster pocket dictionary defines, Thermodynamics: physics of heat.) Historically, it grew out of efforts to construct more efficient heat engines | devices for ex- tracting useful work from expanding hot gases (http://www.answers.com/thermodynamics).
  • Module P7.4 Specific Heat, Latent Heat and Entropy

    Module P7.4 Specific Heat, Latent Heat and Entropy

    FLEXIBLE LEARNING APPROACH TO PHYSICS Module P7.4 Specific heat, latent heat and entropy 1 Opening items 4 PVT-surfaces and changes of phase 1.1 Module introduction 4.1 The critical point 1.2 Fast track questions 4.2 The triple point 1.3 Ready to study? 4.3 The Clausius–Clapeyron equation 2 Heating solids and liquids 5 Entropy and the second law of thermodynamics 2.1 Heat, work and internal energy 5.1 The second law of thermodynamics 2.2 Changes of temperature: specific heat 5.2 Entropy: a function of state 2.3 Changes of phase: latent heat 5.3 The principle of entropy increase 2.4 Measuring specific heats and latent heats 5.4 The irreversibility of nature 3 Heating gases 6 Closing items 3.1 Ideal gases 6.1 Module summary 3.2 Principal specific heats: monatomic ideal gases 6.2 Achievements 3.3 Principal specific heats: other gases 6.3 Exit test 3.4 Isothermal and adiabatic processes Exit module FLAP P7.4 Specific heat, latent heat and entropy COPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1 1 Opening items 1.1 Module introduction What happens when a substance is heated? Its temperature may rise; it may melt or evaporate; it may expand and do work1—1the net effect of the heating depends on the conditions under which the heating takes place. In this module we discuss the heating of solids, liquids and gases under a variety of conditions. We also look more generally at the problem of converting heat into useful work, and the related issue of the irreversibility of many natural processes.
  • Thermodynamic Potentials

    Thermodynamic Potentials

    THERMODYNAMIC POTENTIALS, PHASE TRANSITION AND LOW TEMPERATURE PHYSICS Syllabus: Unit II - Thermodynamic potentials : Internal Energy; Enthalpy; Helmholtz free energy; Gibbs free energy and their significance; Maxwell's thermodynamic relations (using thermodynamic potentials) and their significance; TdS relations; Energy equations and Heat Capacity equations; Third law of thermodynamics (Nernst Heat theorem) Thermodynamics deals with the conversion of heat energy to other forms of energy or vice versa in general. A thermodynamic system is the quantity of matter under study which is in general macroscopic in nature. Examples: Gas, vapour, vapour in contact with liquid etc.. Thermodynamic stateor condition of a system is one which is described by the macroscopic physical quantities like pressure (P), volume (V), temperature (T) and entropy (S). The physical quantities like P, V, T and S are called thermodynamic variables. Any two of these variables are independent variables and the rest are dependent variables. A general relation between the thermodynamic variables is called equation of state. The relation between the variables that describe a thermodynamic system are given by first and second law of thermodynamics. According to first law of thermodynamics, when a substance absorbs an amount of heat dQ at constant pressure P, its internal energy increases by dU and the substance does work dW by increase in its volume by dV. Mathematically it is represented by 풅푸 = 풅푼 + 풅푾 = 풅푼 + 푷 풅푽….(1) If a substance absorbs an amount of heat dQ at a temperature T and if all changes that take place are perfectly reversible, then the change in entropy from the second 풅푸 law of thermodynamics is 풅푺 = or 풅푸 = 푻 풅푺….(2) 푻 From equations (1) and (2), 푻 풅푺 = 풅푼 + 푷 풅푽 This is the basic equation that connects the first and second laws of thermodynamics.
  • Equation of State - Wikipedia, the Free Encyclopedia 頁 1 / 8

    Equation of State - Wikipedia, the Free Encyclopedia 頁 1 / 8

    Equation of state - Wikipedia, the free encyclopedia 頁 1 / 8 Equation of state From Wikipedia, the free encyclopedia In physics and thermodynamics, an equation of state is a relation between state variables.[1] More specifically, an equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions. It is a constitutive equation which provides a mathematical relationship between two or more state functions associated with the matter, such as its temperature, pressure, volume, or internal energy. Equations of state are useful in describing the properties of fluids, mixtures of fluids, solids, and even the interior of stars. Thermodynamics Contents ■ 1 Overview ■ 2Historical ■ 2.1 Boyle's law (1662) ■ 2.2 Charles's law or Law of Charles and Gay-Lussac (1787) Branches ■ 2.3 Dalton's law of partial pressures (1801) ■ 2.4 The ideal gas law (1834) Classical · Statistical · Chemical ■ 2.5 Van der Waals equation of state (1873) Equilibrium / Non-equilibrium ■ 3 Major equations of state Thermofluids ■ 3.1 Classical ideal gas law Laws ■ 4 Cubic equations of state Zeroth · First · Second · Third ■ 4.1 Van der Waals equation of state ■ 4.2 Redlich–Kwong equation of state Systems ■ 4.3 Soave modification of Redlich-Kwong State: ■ 4.4 Peng–Robinson equation of state Equation of state ■ 4.5 Peng-Robinson-Stryjek-Vera equations of state Ideal gas · Real gas ■ 4.5.1 PRSV1 Phase of matter · Equilibrium ■ 4.5.2 PRSV2 Control volume · Instruments ■ 4.6 Elliott, Suresh, Donohue equation of state Processes:
  • Use of Equations of State and Equation of State Software Packages

    Use of Equations of State and Equation of State Software Packages

    USE OF EQUATIONS OF STATE AND EQUATION OF STATE SOFTWARE PACKAGES Adam G. Hawley Darin L. George Southwest Research Institute 6220 Culebra Road San Antonio, TX 78238 Introduction Peng-Robinson (PR) Determination of fluid properties and phase The Peng-Robison (PR) EOS (Peng and Robinson, conditions of hydrocarbon mixtures is critical 1976) is referred to as a cubic equation of state, for accurate hydrocarbon measurement, because the basic equations can be rewritten as cubic representative sampling, and overall pipeline polynomials in specific volume. The Peng Robison operation. Fluid properties such as EOS is derived from the basic ideal gas law along with other corrections, to account for the behavior of compressibility and density are critical for flow a “real” gas. The Peng Robison EOS is very measurement and determination of the versatile and can be used to determine properties hydrocarbon due point is important to verify such as density, compressibility, and sound speed. that heavier hydrocarbons will not condense out The Peng Robison EOS can also be used to of a gas mixture in changing process conditions. determine phase boundaries and the phase conditions of hydrocarbon mixtures. In the oil and gas industry, equations of state (EOS) are typically used to determine the Soave-Redlich-Kwong (SRK) properties and the phase conditions of hydrocarbon mixtures. Equations of state are The Soave-Redlich-Kwaon (SRK) EOS (Soave, 1972) is a cubic equation of state, similar to the Peng mathematical correlations that relate properties Robison EOS. The main difference between the of hydrocarbons to pressure, temperature, and SRK and Peng Robison EOS is the different sets of fluid composition.
  • PHYS 521: Statistical Mechanics Homework #1

    PHYS 521: Statistical Mechanics Homework #1

    PHYS 521: Statistical Mechanics Homework #1 Prakash Gautam April 17, 2018 1. A particular system obeys two equations of state 3As2 As3 T = (thermal equation of state);P = (mechanical equation of state): v v2 Where A is a constant. (a) Find µ as a function of s and v, and then find the fundamental equation. Solution: Given P and T the differential of each of them can be calculated as 6As 3As2 6As2 3As3 dT = ds − dv ) sdT = ds − dv v v2 v v2 3As2 2As3 3As2 2As3 dP = ds − dv ) vdP = ds − dv v2 v3 v v2 The Gibbs-Duhem relation in energy representation allows to calculate the value of µ. dµ = vdP − sdT 3As2 2As3 6As2 3As3 = ds − dv − ds + dv v[ v2 ] v ( v)2 3As2 As3 As3 = − ds − dv = −d v v2 v As3 This can be identified as the total derivative of v so the rel ( ) As3 As3 dµ = −d ) µ = − + k v v Where k is arbitrary constant. We can plug this back to Euler relation to find the fundamental equation as. u = T s − P v + µ 3As3 As3 As3 As3 = − − + k = + k v v v v As3 □ So the fundamental equation of the system is v + k. (b) Find the fundamental equation of this system by direct integration of the molar form of the equation. Solution: The differential form of internal energy is du = T ds − P dv 3As2 As3 = ds − dv v v2 1 As3 As before this is just the total differential of v so the relation leads to ( ) As3 As3 du = d ) u = + k v2 v This k should be the same arbitrary constant that we got in the previous problem.
  • Thermal and Statistical Physics (Part II) Examples Sheet 1 ————————————————————————————

    Thermal and Statistical Physics (Part II) Examples Sheet 1 ————————————————————————————

    Michaelmas 2010 Thermal and Statistical Physics (Part II) Examples Sheet 1 ————————————————————————————- 1. Show that, for a van der Waals gas, the specific heat at constant volume, CV , obeys ∂C V = 0. ∂V T 2. The Gibbs free energy of an imperfect gas containing N molecules is given, in terms of its natural variables T , p and N, by p G = NkBT ln − NA(T ) p, p0 where p0 is a constant and A is a function of T only. Derive expressions in terms of T , p, V , and N for: (a) the equation of state of the gas; (b) the entropy, S; (c) the enthalpy, H; (d) the internal energy, U; (e) the Helmholtz free energy, F . Can all equilibrium thermodynamic information about the gas be obtained from a knowledge of: (i) F (T, V, N); (ii) the equation of state and U(T, p, N)? 3. The entropy of a monatomic ideal gas is given by the Sackur-Tetrode equation which can be written in the form: 3 2 V U / S(U, V, N)= Nk ln α , B N N ( ) where α is a constant to be derived later in the course. Invert this expression to get U(S, V, N). From this, obtain the equation of state expressing p as a function of V, N and T . 4. Use a Maxwell relation and the chain rule to show that for any substance the rate of change of T with p in a reversible adiabatic compression is given by ∂T T ∂V = . ∂p C ∂T S p p Find an equivalent expression for the adiabatic rate of change of T with V , and check that both results are valid for an ideal monatomic gas.
  • Thermodynamically Consistent Modeling and Simulation of Multi-Component Two-Phase Flow Model with Partial Miscibility

    Thermodynamically Consistent Modeling and Simulation of Multi-Component Two-Phase Flow Model with Partial Miscibility

    THERMODYNAMICALLY CONSISTENT MODELING AND SIMULATION OF MULTI-COMPONENT TWO-PHASE FLOW MODEL WITH PARTIAL MISCIBILITY∗ JISHENG KOU† AND SHUYU SUN‡ Abstract. A general diffuse interface model with a realistic equation of state (e.g. Peng- Robinson equation of state) is proposed to describe the multi-component two-phase fluid flow based on the principles of the NVT-based framework which is a latest alternative over the NPT-based framework to model the realistic fluids. The proposed model uses the Helmholtz free energy rather than Gibbs free energy in the NPT-based framework. Different from the classical routines, we combine the first law of thermodynamics and related thermodynamical relations to derive the entropy balance equation, and then we derive a transport equation of the Helmholtz free energy density. Furthermore, by using the second law of thermodynamics, we derive a set of unified equations for both interfaces and bulk phases that can describe the partial miscibility of two fluids. A relation between the pressure gradient and chemical potential gradients is established, and this relation leads to a new formulation of the momentum balance equation, which demonstrates that chemical potential gradients become the primary driving force of fluid motion. Moreover, we prove that the proposed model satisfies the total (free) energy dissipation with time. For numerical simulation of the proposed model, the key difficulties result from the strong nonlinearity of Helmholtz free energy density and tight coupling relations between molar densities and velocity. To resolve these problems, we propose a novel convex-concave splitting of Helmholtz free energy density and deal well with the coupling relations between molar densities and velocity through very careful physical observations with a mathematical rigor.
  • Maxwell Relations

    Maxwell Relations

    MAXWELLRELATIONS 7.I THE MAXWELL RELATIONS In Section 3.6 we observedthat quantities such as the isothermal compressibility, the coefficientof thermal expansion,and the molar heat capacities describe properties of physical interest. Each of these is essentiallya derivativeQx/0Y)r.r..,. in which the variablesare either extensiveor intensive thermodynaini'cparameters. with a wide range of extensive and intensive parametersfrom which to choose, in general systems,the numberof suchpossible derivatives is immense.But thire are azu azu ASAV AVAS (7.1) -tI aP\ | AT\ as),.*,.*,,...: \av)s.N,.&, (7.2) This relation is the prototype of a whole class of similar equalities known as the Maxwell relations. These relations arise from the equality of the mixed partial derivatives of the fundamental relation expressed in any of the various possible alternative representations. 181 182 Maxwell Relations Given a particular thermodynamicpotential, expressedin terms of its (l + 1) natural variables,there are t(t + I)/2 separatepairs of mixed second derivatives.Thus each potential yields t(t + l)/2 Maxwell rela- tions. For a single-componentsimple systemthe internal energyis a function : of three variables(t 2), and the three [: (2 . 3)/2] -arUTaSpurs of mixed second derivatives are A2U/ASAV : AzU/AV AS, AN : A2U / AN AS, and 02(J / AVA N : A2U / AN 0V. Thecompleteser of Maxwell relations for a single-componentsimple systemis given in the following listing, in which the first column states the potential from which the ar\ U S,V I -|r| aP\ \ M)',*: ,s/,,' (7.3) dU: TdS- PdV + p"dN ,S,N (!t\ (4\ (7.4) \ dNls.r, \ ds/r,r,o V,N _/ii\ (!L\ (7.5) \0Nls.v \0vls.x UITI: F T,V ('*),,.: (!t\ (7.6) \0Tlv.p dF: -SdT - PdV + p,dN T,N -\/ as\ a*) ,.,: (!"*),.
  • Thermodynamics.Pdf

    Thermodynamics.Pdf

    1 Statistical Thermodynamics Professor Dmitry Garanin Thermodynamics February 24, 2021 I. PREFACE The course of Statistical Thermodynamics consist of two parts: Thermodynamics and Statistical Physics. These both branches of physics deal with systems of a large number of particles (atoms, molecules, etc.) at equilibrium. 3 19 One cm of an ideal gas under normal conditions contains NL = 2:69×10 atoms, the so-called Loschmidt number. Although one may describe the motion of the atoms with the help of Newton's equations, direct solution of such a large number of differential equations is impossible. On the other hand, one does not need the too detailed information about the motion of the individual particles, the microscopic behavior of the system. One is rather interested in the macroscopic quantities, such as the pressure P . Pressure in gases is due to the bombardment of the walls of the container by the flying atoms of the contained gas. It does not exist if there are only a few gas molecules. Macroscopic quantities such as pressure arise only in systems of a large number of particles. Both thermodynamics and statistical physics study macroscopic quantities and relations between them. Some macroscopics quantities, such as temperature and entropy, are non-mechanical. Equilibruim, or thermodynamic equilibrium, is the state of the system that is achieved after some time after time-dependent forces acting on the system have been switched off. One can say that the system approaches the equilibrium, if undisturbed. Again, thermodynamic equilibrium arises solely in macroscopic systems. There is no thermodynamic equilibrium in a system of a few particles that are moving according to the Newton's law.
  • Equation of State

    Equation of State

    EQUATION OF STATE Consider elementary cell in a phase space with a volume 3 ∆x ∆y ∆z ∆px ∆py ∆pz = h , (st.1) where h =6.63×10−27 erg s is the Planck constant, ∆x ∆y ∆z is volume in ordinary space measured 3 −1 3 in cm , and ∆px ∆py ∆pz is volume in momentum space measured in ( g cm s ) . According to quantum mechanics there is enough room for approximately one particle of any kind within any elementary cell. More precisely, an average number of particles per cell is given as g n = , (st.2) av e(E−µ)/kT ± 1 with a ”+” sign for fermions, and a ”−” sign for bosons. The corresponding distributions are called Fermi-Dirac and Bose-Einstein, respectively. Particles with a spin of 1/2 are called fermions, while those with a spin 0, 1, 2... are called bosons. Electrons and protons are fermions, photons are bosons, while larger nuclei or atoms may be either fermions or bosons, depending on the total spin of such − a composite particle. In the equation (st.2) E is the particle energy, k = 1.38 × 10−16 erg K 1 is the Boltzman constant, T is temperature, µ is chemical potential, and g is a number of different quantum states a particle may have within the cell. The meaning of temperature is obvious, while chemical potential will become more familiar later on. In most cases it will be close to the rest mass of a particle under consideration. If there are anti-particles present in equilibrium with particles, and particles have chemical potential µ then antiparticles have chemical potential µ − 2m, where m is their rest mass.