18.13 Review & Summary

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1 5/2/10 10:04 PM Print this page Review & Summary Temperature; Thermometers Temperature is an SI base quantity related to our sense of hot and cold. It is measured with a thermometer, which contains a working substance with a measurable property, such as length or pressure, that changes in a regular way as the substance becomes hotter or colder. Zeroth Law of Thermodynamics When a thermometer and some other object are placed in contact with each other, they eventually reach thermal equilibrium. The reading of the thermometer is then taken to be the temperature of the other object. The process provides consistent and useful temperature measurements because of the zeroth law of thermodynamics: If bodies A and B are each in thermal equilibrium with a third body C (the thermometer), then A and B are in thermal equilibrium with each other. The Kelvin Temperature Scale In the SI system, temperature is measured on the Kelvin scale, which is based on the triple point of water ( K). Other temperatures are then defined by use of a constant-volume gas thermometer, in which a sample of gas is maintained at constant volume so its pressure is proportional to its temperature. We define the temperature T as measured with a gas thermometer to be (18-6) Here T is in kelvins, and p 3 and p are the pressures of the gas at K and the measured temperature, respectively. Celsius and Fahrenheit Scales The Celsius temperature scale is defined by (18-7) with T in kelvins. The Fahrenheit temperature scale is defined by (18-8) Thermal Expansion All objects change size with changes in temperature. For a temperature change change in any linear dimension L is given by, a (18-9) in which is the coefficient of linear expansion. The change in the volume V of a solid or liquid is (18-10) Here is the material s coefficient of volume expansion. Heat Heat Q is energy that is transferred between a system and its environment because of a temperature difference between them. It can be measured in joules (J), calories (cal), kilocalories (Cal or kcal), or British thermal units (Btu), with (18-12) Heat Capacity and Specific Heat If heat Q is absorbed by an object, the object s temperature change is related to Q by Page 1 of 3

2 5/2/10 10:04 PM (18-13) in which C is the heat capacity of the object. If the object has mass m, then (18-14) where c is the specific heat of the material making up the object. The molar specific heat of a material is the heat capacity per mole, which means per elementary units of the material. Heat of Transformation Heat absorbed by a material may change the material s physical state for example, from solid to liquid or from liquid to gas. The amount of energy required per unit mass to change the state (but not the temperature) of a particular material is its heat of transformation L. Thus, (18-16) The heat of vaporization L V is the amount of energy per unit mass that must be added to vaporize a liquid or that must be removed to condense a gas. The heat of fusion L F is the amount of energy per unit mass that must be added to melt a solid or that must be removed to freeze a liquid. Work Associated with Volume Change A gas may exchange energy with its surroundings through work. The amount of work W done by a gas as it expands or contracts from an initial volume V i to a final volume V f is given by (18-25) The integration is necessary because the pressure p may vary during the volume change. First Law of Thermodynamics The principle of conservation of energy for a thermodynamic process is expressed in the first law of thermodynamics, which may assume either of the forms (18-26) or (18-27) E int represents the internal energy of the material, which depends only on the material s state (temperature, pressure, and volume). Q represents the energy exchanged as heat between the system and its surroundings; Q is positive if the system absorbs heat and negative if the system loses heat. W is the work done by the system; W is positive if the system expands against an external force from the surroundings and negative if the system contracts because of an external force. Q and W are path dependent; is path independent. Applications of the First Law The first law of thermodynamics finds application in several special cases: Page 2 of 3

3 5/2/10 10:04 PM Conduction, Convection, and Radiation The rate P cond at which energy is conducted through a slab whose faces are maintained at temperatures T H and T C is (18-32) in which A and L are the face area and length of the slab, and k is the thermal conductivity of the material. Convection occurs when temperature differences cause an energy transfer by motion within a fluid. Radiation is an energy transfer via the emission of electromagnetic energy. The rate P rad at which an object emits energy via thermal radiation is (18-38) where is the Stefan Boltzmann constant, is the emissivity of the object s surface, A is its surface area, and T is its surface temperature (in kelvins). The rate P abs at which an object absorbs energy via thermal radiation from its environment, which is at the uniform temperature T env (in kelvins), is (18-39) Copyright 2008 John Wiley & Sons, Inc. All rights reserved. Page 3 of 3

4 5/2/10 10:05 PM Print this page Review & Summary Kinetic Theory of Gases The kinetic theory of gases relates the macroscopic properties of gases (for example, pressure and temperature) to the microscopic properties of gas molecules (for example, speed and kinetic energy). Avogadro s Number One mole of a substance contains N A (Avogadro s number) elementary units (usually atoms or molecules), where N A is found experimentally to be (19-1) One molar mass M of any substance is the mass of one mole of the substance. It is related to the mass m of the individual molecules of the substance by (19-4) The number of moles n contained in a sample of mass M sam, consisting of N molecules, is given by (19-2, 19-3) Ideal Gas An ideal gas is one for which the pressure p, volume V, and temperature T are related by (19-5) Here n is the number of moles of the gas present and R is a constant (8.31 J/mol K) called the gas constant. The ideal gas law can also be written as (19-9) where the Boltzmann constant k is (19-7) Work in an Isothermal Volume Change The work done by an ideal gas during an isothermal (constanttemperature) change from volume V i to volume V f is (19-14) Pressure, Temperature, and Molecular Speed The pressure exerted by n moles of an ideal gas, in terms of the speed of its molecules, is (19-21) where is the root-mean-square speed of the molecules of the gas. With Eq this gives Page 1 of 3

5 5/2/10 10:05 PM (19-22) Temperature and Kinetic Energy The average translational kinetic energy K avg per molecule of an ideal gas is (19-24) Mean Free Path The mean free path given by of a gas molecule is its average path length between collisions and is (19-25) where N/V is the number of molecules per unit volume and d is the molecular diameter. Maxwell Speed Distribution The Maxwell speed distribution P(v) is a function such that P(v) dv gives the fraction of molecules with speeds in the interval dv at speed v: (19-27) Three measures of the distribution of speeds among the molecules of a gas are (19-31) (19-35) and the rms speed defined above in Eq Molar Specific Heats The molar specific heat C V of a gas at constant volume is defined as (19-39, 19-41) in which Q is the energy transferred as heat to or from a sample of n moles of the gas, is the resulting temperature change of the gas, and is the resulting change in the internal energy of the gas. For an ideal monatomic gas, (19-43) The molar specific heat C p of a gas at constant pressure is defined to be (19-46) in which Q, n, and are defined as above. C p is also given by (19-49) For n moles of an ideal gas, (19-44) Page 2 of 3

6 5/2/10 10:05 PM If n moles of a confined ideal gas undergo a temperature change internal energy of the gas is due to any process, the change in the (19-45) in which the appropriate value of C V must be substituted, according to the type of ideal gas. Degrees of Freedom and C V We find C V by using the equipartition of energy theorem, which states that every degree of freedom of a molecule (that is, every independent way it can store energy) has associated with it on average an energy per molecule ( per mole). If f is the number of degrees of freedom, then and (19-51) For monatomic gases (three translational degrees); for diatomic gases (three translational and two rotational degrees). Adiabatic Process When an ideal gas undergoes a slow adiabatic volume change (a change for which pressure and volume are related by ), its (19-53) in which constant. is the ratio of molar specific heats for the gas. For a free expansion, however, Copyright 2008 John Wiley & Sons, Inc. All rights reserved. Page 3 of 3

7 5/2/10 10:06 PM Print this page 20.9 Review & Summary One-Way Processes An irreversible process is one that cannot be reversed by means of small changes in the environment. The direction in which an irreversible process proceeds is set by the change in entropy ΔS of the system undergoing the process. Entropy S is a state property (or state function) of the system; that is, it depends only on the state of the system and not on the way in which the system reached that state. The entropy postulate states (in part): If an irreversible process occurs in a closed system, the entropy of the system always increases. Calculating Entropy Change The entropy change for an irreversible process that takes a system from an initial state i to a final state f is exactly equal to the entropy change for any reversible process that takes the system between those same two states. We can compute the latter (but not the former) with (20-1) Here Q is the energy transferred as heat to or from the system during the process, and T is the temperature of the system in kelvins during the process. For a reversible isothermal process, Eq reduces to (20-2) When the temperature change of a system is small relative to the temperature (in kelvins) before and after the process, the entropy change can be approximated as (20-3) where T avg is the system s average temperature during the process. When an ideal gas changes reversibly from an initial state with temperature T i and volume V i to a final state with temperature T f and volume V f, the change in the entropy of the gas is (20-4) The Second Law of Thermodynamics This law, which is an extension of the entropy postulate, states: If a process occurs in a closed system, the entropy of the system increases for irreversible processes and remains constant for reversible processes. It never decreases. In equation form, (20-5) Engines An engine is a device that, operating in a cycle, extracts energy as heat from a hightemperature reservoir and does a certain amount of work. The efficiency of any engine is defined as (20-11) In an ideal engine, all processes are reversible and no wasteful energy transfers occur due to, say, friction and Page 1 of 3

8 5/2/10 10:06 PM turbulence. A Carnot engine is an ideal engine that follows the cycle of Fig Its efficiency is (20-12,20-13) in which T H and T L are the temperatures of the high- and low-temperature reservoirs, respectively. Real engines always have an efficiency lower than that given by Eq Ideal engines that are not Carnot engines also have lower efficiencies. A perfect engine is an imaginary engine in which energy extracted as heat from the high-temperature reservoir is converted completely to work. Such an engine would violate the second law of thermodynamics, which can be restated as follows: No series of processes is possible whose sole result is the absorption of energy as heat from a thermal reservoir and the complete conversion of this energy to work. Refrigerators A refrigerator is a device that, operating in a cycle, has work W done on it as it extracts energy as heat from a low-temperature reservoir. The coefficient of performance K of a refrigerator is defined as (20-14) A Carnot refrigerator is a Carnot engine operating in reverse. For a Carnot refrigerator, Eq becomes (20-15,20-16) A perfect refrigerator is an imaginary refrigerator in which energy extracted as heat from the low-temperature reservoir is converted completely to heat discharged to the high-temperature reservoir, without any need for work. Such a refrigerator would violate the second law of thermodynamics, which can be restated as follows: No series of processes is possible whose sole result is the transfer of energy as heat from a reservoir at a given temperature to a reservoir at a higher temperature. Entropy from a Statistical View The entropy of a system can be defined in terms of the possible distributions of its molecules. For identical molecules, each possible distribution of molecules is called a microstate of the system. All equivalent microstates are grouped into a configuration of the system. The number of microstates in a configuration is the multiplicity W of the configuration. For a system of N molecules that may be distributed between the two halves of a box, the multiplicity is given by (20-20) in which n 1 is the number of molecules in one half of the box and n 2 is the number in the other half A basic assumption of statistical mechanics is that all the microstates are equally probable. Thus, configurations with a large multiplicity occur most often. When N is very large (say, molecules or more), the molecules are nearly always in the configuration in which. The multiplicity W of a configuration of a system and the entropy S of the system in that configuration are related by Boltzmann s entropy equation: (20-21) where is the Boltzmann constant. When N is very large (the usual case), we can approximate ln N! with Stirling s approximation: Page 2 of 3

9 5/2/10 10:06 PM (20-22) Copyright 2008 John Wiley & Sons, Inc. All rights reserved. Page 3 of 3