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1 1 Macroscopic Point of View Separation of a restricted region of space or a finite portion of matter from its surroundings means of a closed surface called the boundary The region within the arbitrary boundary and on which the attention is focused is called the system Everything outside the system...

Description

1 1.1 Macroscopic Point of View • • • • • • • •

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Separation of a restricted region of space or a finite portion of matter from its surroundings by means of a closed surface called the boundary The region within the arbitrary boundary and on which the attention is focused is called the system Everything outside the system that has a direct bearing on the system’s behavior is known as the surroundings, which could be another system If no matter crosses the boundary, then the system is closed If there is an exchange of matter between system and surroundings, then the system is open Macroscopic point of view considers variables or characteristics of a system at approximately the human scale, or larger Microscopic point of view considers variables or characteristics of a system at approximately the molecular scale, or smaller Quantities of mass, composition, volume, pressure, and temperature refer to the large-scale characteristics, or aggregate properties, of the system and provide a macroscopic description. The quantities are, therefore, called macroscopic coordinates. Macroscopic coordinates, in general have the following properties in common: o They involve no special assumptions concerning the structure of matter, fields, or radiation o They are few in number needed to describe the system o They are fundamental, as suggested more or less directly by our sensory perceptions o They can, in general, be directly measured A macroscopic description of a system involves the specification of a few fundamental measurable properties of a system Thermodynamics is the branch of natural science that deals with the macroscopic properties of characteristics of nature and always includes the macroscopic coordinates of temperature for every system

1.2 Microscopic Point of View •

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A system is considered to consist of an enormous number N of particles, each of which is capable of existing in a set of states whose energies are 𝜖1 , 𝜖2 , … . The particles are assumed to interact with one another by means of collisions or by forces caused by fields. The system of particles may be imagined to be isolated or, in some cases, may be considered to be embedded in a set of similar systems, or ensemble of systems. The fundamental problem is to find the number of particles in each of the microscopic energy states (known as the populations of the states) when equilibrium is reached Statistical mechanics is the branch of natural science that deals with the microscopic characteristics of nature A microscopic description of a system involves the following properties: o Assumptions are made concerning the structure of matter, fields, or radiation o Many quantities must be specified to describe the system o These quantities specified are not usually suggested by our sensory perceptions, but rather by our mathematical models o They cannot be directly measured, but must be calculated A microscopic description of a system involves various assumptions about the internal structure of the system and then calculations of system-wide characteristics

1.3 Macroscopic vs. Microscopic Points of View •

The few directly measurable properties whose specification constitutes the macroscopic description are really averages, over a period of time, of a large number of molecular collisions made on a unit of area

2 Example: Pressure is the average rate of change of linear momentum due to the large number of molecular collisions made on a unit of area. Pressure, however, is a property that is perceived by our senses. The microscopic point of view goes much further than our senses and many direct experiments. It assumes the structure of microscopic particles, their motion, their energy states, their interactions, etc., and then calculates measurable quantities When we seek to understand the physical reality of a result of a microscopic calculation, we look to the macroscopic point of view for guidance o

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1.4 Scope of Thermodynamics • • • • •

The attention is directed to the interior of a system A macroscopic point of view is adopted, and emphasis is placed on those macroscopic quantities which have a bearing on the internal state of a system Macroscopic quantities, including temperature, having a bearing on the internal state of the system are called thermodynamic coordinates. Such coordinates serve to determine the internal energy of a system. It is the purpose of thermodynamics to find, among the thermodynamic coordinates, general relations that are consistent with the fundamental laws of thermodynamics A system that may be described in terms of thermodynamic coordinates is called a thermodynamic system

1.5 Thermal Equilibrium and the Zeroth Law • •

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• •

Pressure and volume are independent coordinates but are related by Boyle’s law In referring to any unspecified system, we shall use the symbols X and Y for the pair of independent coordinates, where X refers to a generalized force (ex. pressure of a gas) and Y refers to a generalized displacement (ex. volume of a gas) A state of system in which the coordinates X and Y have definite values that remain constant so long as the external conditions are unchanged is called an equilibrium state If a wall is adiabatic, an equilibrium state for system A may coexist with any equilibrium state of system B for all attainable values of the four quantities, X, Y, X’, and Y’ – provided only that the wall is able to withstand the stress associated with the difference between the two sets of coordinates o An adiabatic wall prevents two systems from communicating with each other and coming to thermal equilibrium with each other. IT DOES NOT ALLOW FOR EXCHANGE OF HEAT If the two systems are separate by a diathermic wall, the values of X, Y, X’, and Y’ will change spontaneously until an equilibrium state of the combined system is attained. The two systems are then said to be in thermal equilibrium with each other

Thermal equilibrium is the state achieved by two (or more) systems, characterized by restricted values of the coordinates of the systems, after they have been in communication with each other through a diathermic wall. Zeroth law of thermodynamics: two systems in thermal equilibrium with a third are in thermal equilibrium w/ each other

3 1.6 Concept of Temperature • • •

Temperature is a measure of the hotness of a given macroscopic object, as felt by the human body An isotherm is the locus of all points representing states in which a system is in thermal equilibrium with one state of another system The temperature of a system is a property that determines whether or not a system is in thermal equilibrium with other systems

1.7 Thermometers and Measurement of Temperature •

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To establish an empirical temperature scale, we select some system with coordinates X and Y as a standard, which we call a thermometer, and adopt a set of rules for assigning a numerical value to the temperature associated with each of its isotherms. To every other system in thermal equilibrium with the thermometer, we assign the same number for the temperature The simplest procedure is to choose any convenient path in the X – Y plane which intersects the isotherms at points, each of which has the same Y-coordinate but a different X-coordinate. The temperature associated with each isotherm is then taken to be a convenient function of the X at this intersection point The coordinate X is called the thermometric property, and the form of the thermometric function 𝜃(𝑋) determines the empirical temperature scale Let X stand for any one of the thermometric properties listed in Table 1.1 (page 33), and let us decide arbitrarily to define the temperature scale so that the empirical temperature θ is directly proportional to X The temperature common to the thermometer and to all systems in thermal equilibrium with it can be given by: 𝜃(𝑋) = 𝑎𝑋 *constant Y and arbitrary constant a* …(1.1) This equation applies, in general, to a thermometer placed in contact with a system whose temperature 𝜃(𝑋) is to be measured. Therefore, it applies when the thermometer is placed in contact with an arbitrarily chosen standard system in a reproducible state; such a state of an arbitrarily chosen standard system is called a fixed point, that is, fixed temperature o The fixed point provides a reference temperature for the determination of temperature scales The state in which ice, liquid water, and water vapor coexist in equilibrium, a state known as the triple point of water, provides the standard reference temperature → 273.16 K → 0.01°C 𝑎=

•

273.16𝐾 𝑋𝑇𝑃

In view of equation 1.2, equation 1.1 may be rewritten: 𝑋

𝜃(𝑋) = 273.16𝐾 𝑋 •

…(1.2)

𝑇𝑃

*constant Y*

…(1.3)

The temperature of the triple point of water is the standard fixed point of thermometry

1.10 Ideal-Gas Temperature • • •

Ideal-gas law: 𝑃𝑉 = 𝑛𝑅𝑇 …(1.5) Θ indicates the real-gas temperature and T the thermodynamic ideal-gas temperature Applying equation 1.5 initially to the gas at the assigned temperature of 273.16K and then to the gas at the unknown empirical temperature, one obtains the proportion: 𝑃 𝑃𝑇𝑃

• •

=

𝜃 273.16𝐾

𝜃 = 273.16𝐾

or

𝑃 𝑃𝑇𝑃

*constant V*

All gases indicate the same temperature as PTP is lowered and made to approach zero We define the ideal-gas temperature T by the equation: 𝑇 = 273.16𝐾 lim

𝑃

𝑃𝑇𝑃 →0 𝑃𝑇𝑃

*constant V*

…(1.7)

…(1.6)

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The temperature T = 0 remains as yet undefined by means of thermometry In the temperature region in which a gas thermometer may be used, the ideal-gas scale and the Kelvin thermodynamic scale are identical The statement that at absolute zero all atomic motion ceases is erroneous o Such a statement involves an assumption connecting the purely macroscopic concept of temperature and the microscopic concept of atomic motion. If we want thermodynamics to be general, this is precisely the sort of assumption that must be avoided o When it is necessary in statistical mechanics to correlate temperature to atomic or molecular motion, it is found that classical statistical mechanics must be modified with the aid of quantum mechanics and that, when this modification is carried out, the particles of a substance at absolute zero have a finite amount of residual vibrational energy, known as the zero-point energy

2.1 Thermodynamic Equilibrium • • • •

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When the macroscopic coordinates change in any way whatsoever, either spontaneously or by virtue of outside influence, the system is said to undergo a change of state When a system is not influenced in any way by its surroundings, it is said to be isolated. This system is of little importance in practical applications of thermodynamics When there is no unbalanced force or torque in the interior of a system ad also none between a system and its surroundings, the system is in mechanical equilibrium When a system in mechanical equilibrium does not tend to undergo a spontaneous change of internal structure (ex. a chemical reaction) or a transfer of matter from one part of the system to another (ex. diffusion or solution), however slow, then it is in chemical equilibrium Thermal equilibrium exists when there is no spontaneous change in the coordinates of a system in mechanical and chemical equilibrium when it is separated from its surrounding by diathermic walls. There is no exchange of heat between the system and its surroundings o All parts of a system are at the same temperature, and this temperature is the same as that of the surroundings When the conditions for all three types of equilibrium are satisfied, the system is said to be in thermodynamic equilibrium o It is apparent that there will be no tendency whatever for any change of state, either of the system or of the surroundings, to occur States of thermodynamic equilibrium can be described in terms of macroscopic coordinates that do not involve the time, that is, in terms of thermodynamic coordinates When the conditions for any one of the three types of equilibrium that constitute thermodynamic equilibrium are not satisfied, the system is in a nonequilibrium state When the conditions for mechanical and thermal equilibrium are not satisfied, the states traversed by a system cannot be described in terms of thermodynamic coordinates referring to the system as a whole

2.2 Equation of State • •

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Of the three thermodynamic coordinates P, V, and T, only two are independent variables There exists an equation of equilibrium which connects the thermodynamic coordinates and which robs one of them of its independence o An equation of state is a mathematical function relating the appropriate thermodynamic coordinates of a system in equilibrium Example: ideal-gas law o 𝑃𝑉 = 𝑛𝑅𝑇 …(2.1) → 𝑃𝑣 = 𝑅𝑇, where v = V/n

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van der Waals equation takes into account particle interactions and the finite size of the particles 𝐴

(𝑃 + 𝑣2) (𝑣 − 𝑏) = 𝑅𝑇, where a and b are constants appropriate to the specific gas

…(2.2)

2.3 Hydrostatic Systems • •

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Any isotropic system of constant mass and constant composition that exerts on the surroundings a uniform hydrostatic pressure, in the absence of gravitational, electric, and magnetic effects is a hydrostatic system These systems are divided into the following categories: o A pure substance, which is a single chemical compound in the form of a solid, a liquid, a gas, a mixture of any two, or a mixture of all three o A homogeneous mixture of different compounds, such as a mixture of inert gases, a mixture of chemically active gases, a mixture of liquids, or a solution o A heterogeneous mixture, such as a mixture of different gases in contact w/a mixture of different liquids Pressure is measured in the SI unit of pascal (N/m2), volume in m3, and the temperature in kelvin Every infinitesimal change in thermodynamics must satisfy the requirement that it represents a change in a quantity which is small with respect to the quantity itself and large in comparison with the effect produced by the behavior of a few molecules Example: V is a function of T and P → V = V(T,P) o

𝜕𝑉

𝜕𝑉

𝑑𝑉 = ( ) 𝑑𝑇 + ( 𝜕𝑃) 𝑑𝑃 𝜕𝑇 𝑇 1 𝜕𝑉

𝑃

𝛽 = 𝑉 (𝜕𝑇)

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Volume expansivity:

•

Isothermal compressibility:

𝑃

…(2.3) 1 𝜕𝑉

𝜅 = − ( 𝜕𝑃) 𝑉

…(2.4)

𝑇

2.4 Mathematical Theorems 𝜕𝑃

𝛽 𝜅

𝛽

𝑑𝑃 = 𝜅 𝑑𝑇

•

(

•

If temperature changes a finite amount from Ti to Tf at constant V, the pressure will change from Pi to Pf

) =

𝜕𝑇 𝑉

o

→ 𝑇 𝛽

𝑃𝑓 − 𝑃𝑖 = ∫𝑇 𝑓 𝜅 𝑑𝑇

𝛽

𝑃𝑓 − 𝑃𝑖 = 𝜅 (𝑇𝑓 − 𝑇𝑖 )

→

𝑖

2.5 Stretched Wire •

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A sufficiently complete thermodynamic description of a wire is given in terms of only three coordinates: o The tension in the wire τ, measured in newtons (N) o The length of the wire L, measured in meters (m) o The absolute temperature T, measured in kelvin (K) For a wire at constant temperature within the limit of elasticity, Hooke’s law applies for the tension τ in a stretched wire: 𝜏 = −𝑘(𝐿 − 𝐿0 ) *k = Hooke’s constant, L0 = length at zero tension* …(2.8) 1 𝜕𝐿

𝛼 = (𝜕𝑇) 𝐿

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Linear expansivity:

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Isothermal Young’s modulus:

𝜏

…(2.9) 𝐿 𝜕𝜏

𝑌 = 𝐴 ( 𝜕𝐿)

𝑇

3.1 Work • • •

Work is done if a system undergoes a displacement under the action of a force If a system as a whole exerts a force on its surroundings and a displacement takes place, the work that is done either by the system or on the system is external work The work done by one part of a system on another part is internal work *not discussed in macroscopic thermo*

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When work is done on the system, work is positive (contraction) When work is done by the system, work is negative (expansion)

3.2 Quasi-static Process • •

Ideal situation: external forces acting on a system are varied only slightly so that the unbalanced force is infinitesimal, and the process proceeds infinitesimally slowly During a quasi-static process, the system is at all times infinitesimally near a state of thermodynamic equilibrium

3.3 Work in Changing the Volume of a Hydrostatic System •

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𝑑𝑊 = −𝑃𝑑𝑉 …(3.1) o The presence of the minus sign before PdV ensures that a negative dV (a compression) gives rise to positive work done on the system and, conversely, a positive dV (an expansion) yields negative work done by the system, which is based on the convention that work done on a system increases the energy of the system Unit of work: joule (J) In a finite quasi-static process in which the volume changes from Vi to Vf, the amount of work W done by the system is: 𝑉

𝑊 = − ∫𝑉 𝑓 𝑃 𝑑𝑉 𝑖

…(3.2)

3.4 PV Diagram

3.5 Hydrostatic Work Depends on the Path • •

Isobaric: constant pressure, Isochoric: constant volume, Isothermal: constant temperature The work done by a system depends not only on the initial and final states but also on the intermediate states, namely, on the path of integration o

𝑉

For a quasi-static process, 𝑊 = − ∫𝑉 𝑓 𝑃 𝑑𝑉 cannot be integrated until P is specified as a function of V 𝑖

using an appropriate equation of state 3.6 Calculation of ∫ P dV for Quasi-static Processes •

Quasi-static isothermal expansion or compression of an ideal gas

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𝑉

o

Work is given by 𝑊 = − ∫𝑉 𝑓 𝑃 𝑑𝑉

o

Equation of state for ideal gas: PV = nRT → P = nRT/V

o

𝑊 = − ∫𝑉 𝑓

𝑖

𝑉 𝑛𝑅𝑇 𝑉 𝑖

𝑑𝑉

𝑉 𝑑𝑉 𝑖 𝑉

𝑊 = −𝑛𝑅𝑇 ∫𝑉 𝑓

→

→

𝑊 = −𝑛𝑅𝑇 ln

𝑉𝑓 𝑉𝑖

Quasi-static isothermal increase of pressure on a solid o o o

1

𝜕𝑉

Path of integration is determined by the isothermal compressibility 𝜅 = − 𝑉 (𝜕𝑃) 𝜕𝑉 ( ) 𝜕𝑃 𝑇

𝜕𝑉 𝜕𝑉 𝑑𝑉 = (𝜕𝑃) 𝑑𝑃 + ( 𝜕𝑇) 𝑑𝑇, 𝑝 𝑇 𝑃 𝑊 = − ∫𝑃 𝑓 −𝜅𝑉𝑃 𝑑𝑃 → 𝑖

= −𝜅𝑉 𝑃

𝑊 = 𝜅𝑉 ∫𝑃 𝑓 𝑃 𝑑𝑃 𝑖

→

𝑑𝑉 = −𝜅𝑉 𝑑𝑃

→

𝑊≈

𝜅𝑉 2

𝑇

(𝑃𝑓2 − 𝑃𝑖2 )

3.7 Work in Changing the Length of a Wire • •

If the length of a wire with tension τ is changed an infinitesimal length dL by an external force equal and opposite to the tension, then the infinitesimal amount of work that is done on the wire is 𝑑𝑊 = 𝜏 𝑑𝐿 …(3.3) 𝐿

For a finite change of length: 𝑊 = ∫𝐿 𝑓 𝜏 𝑑𝐿 𝑖

3.8 Work in Changing the Area of Surface Film •

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Consider a soap film, which consists of two surfaces enclosing water, stretched across a wire framework. The right side of the wire framework is movable. If the moveable wire has a length L and the surface tension in one surface is γ, then the external force F exerted on both surfaces is equal in magnitude to 2γL. For an infinitesimal displacement dx, the work is 𝑑𝑊 = 2𝛾𝐿 𝑑𝑥 o For two films, 2𝐿𝑑𝑥 = 𝑑𝐴, so 𝑑𝑊 = 𝛾 𝑑𝐴 𝐴

For a finite change from an initial area to a final area, the expression for work is 𝑊 = ∫ 𝐴 𝑓𝛾 𝑑𝐴 𝑖

4.1 Work and Heat • ...