In practice, several refinements and corrections are necessary. The statement (∂S/∂z)T → 0 as T → 0 is incorporated in another Law: As the temperature of any system approaches the lowest possible temperature of 0 K the entropy of the system assumes a particular, least value when the system is in its lowest energy state. Consider the classical statistical mechanics result for the entropy of an ideal gas, given by the equation (11.81). This quantum violation might not work in practice, though, so the researchers are collaborating with a team of experimentalists to see if it is possible to get around the second law of thermodynamics. Furthermore, what is the behavior of the entropy and other thermodynamic functions as the temperature approaches absolute zero? Hence, for practical purposes, we may ignore this contribution, thus allowing us to set the effective entropy at the absolute zero to zero. We will not attempt to synthesize and reconcile the details here, but present only the main ideas. An elementary form of optical pyrometer. The number of microstates W satisfies the inequalities. According to the third law of thermodynamics, the entropy of a system in internal equilibrium approaches a constant independent of phase as the absolute temperature tends to zero. because it is rarely applied to our day to day life. Exceptions to setting S0 = 0 arise whenever configurational disorder must be taken into account, as in cases where a material may be disordered or be in several states of equal energy that are frozen in at T = 0. ScienceDirect ® is a registered trademark of Elsevier B.V. ScienceDirect ® is a registered trademark of Elsevier B.V. URL: https://www.sciencedirect.com/science/article/pii/B9780128033043000041, URL: https://www.sciencedirect.com/science/article/pii/B9780123738776500034, URL: https://www.sciencedirect.com/science/article/pii/S1049250X15000130, URL: https://www.sciencedirect.com/science/article/pii/B9780120598601500266, URL: https://www.sciencedirect.com/science/article/pii/B978012384956400015X, URL: https://www.sciencedirect.com/science/article/pii/B9780124080904000025, Thermodynamics of Quantum Systems Under Dynamical Control, David Gelbwaser-Klimovsky, ... Gershon Kurizki, in, Advances In Atomic, Molecular, and Optical Physics, Nils Dalarsson, ... Leonardo Golubović, in, Advances in Atomic, Molecular, and Optical Physics, Erik Torrontegui, ... Juan Gonzalo Muga, in, Knowing the transient excitation energy is also important to quantify the principle of unattainability of zero temperature, first enunciated by Nernst. The ideal gas scale coincides with the Kelvin scale of temperature (Section 24.4). Here, I’ll also tell you why entropy of pure crystalline substances is zero.. This principle is usually formulated as the impossibility to reduce the temperature of any system to the absolute zero in a finite number of operations, and identified with the third law of thermodynamics. Steam generators, steam turbines, condensers and feedwater pumps constitute a heat engine, that is subject to the efficiency limitations imposed by the second law of thermodynamics. The Third Law of Thermodynamics says that a perfect crystalline structure at absolute zero temperatures will have zero disorder or entropy. The Third Law of Thermodynamics was first formulated by German chemist and physicist Walther Nernst. Figure 21.2. The entropy reaches this value with zero slope taken with respect to all thermodynamic deformation coordinates. A lower bound Bn for the time-averaged energy of the nth expanding mode En¯ (time averages from 0 to tf will be denoted by a bar) is found by applying calculus of variations (Chen and Muga, 2010), so that En¯≥Bn. Like time or gravity, nothing in the universe is exempt from these laws. (1.18.1) becomes. (e) Materials in which it is important to take account of isotopic distributions or nuclear spin degeneracies, if these quantities change in a given process. The boiling point is the temperature at which a liquid and its vapor are in thermal equilibrium. 21.1 is fixed by assigning a temperature to some standard fixed point. The most fundamental scale of temperature, called the Kelvin scale, is based on the laws of thermodynamics. If you had the opportunity to assign new values to these two fixed points, what values would you choose? The Rankine degree is equal to the Fahrenheit degree, and thus is only 5/9 as large as the Celsius degree and the kelvin, Consequently, a temperature on the Rankine scale is 9/5 as great as the corresponding Kelvin temperature, For example, the Rankine temperature of the ice point of water is. The second law of thermodynamics leads to the definition of entropy and calculation of differences of entropy. With the exception of non-crystalline solids ( glasses) the entropy of a system at absolute zero is typically close to zero. Clearly, it is awkward to have to acknowledge at a later stage that the very existence of such a working substance is denied by the Third Law. The third law of thermodynamics states that as the temperature approaches absolute zero (0 K), the entropy of a system approaches a constant (and minimum) value. A consequence is that tf≥(2n+1)ℏ/(2ωfEn¯). An important application of the third law of thermodynamics is that it helps in the calculation of the absolute entropy of a substance at any temperature ‘T’. Figure 21.1. The corresponding steam-point temperature is. Also, any equilibrium state that remains intrinsically disordered as T → 0 will have a nonzero entropy; examples are furnished in the Remarks section.1. These are the Celsius, Fahrenheit, and Rankine scales. As we shall see in this chapter, both of these questions cannot be addressed using the first two laws alone. First: You can't win. On the Celsius scale, the ice point is 0°C and the steam point is 100°C. The ‘ third law of thermodynamics ’ deals with events as T?0, where d Q / T might diverge. We can insert this result into Eq. Despite the above disclaimer one often does set S0 = 0, namely when So is not altered during a given process, in which case the actual entropy change does not depend on the value assigned to S0. We have seen that entropy is a measure of chaos in a system. George B. Arfken, ... Joseph Priest, in University Physics, 1984. The interested reader is referred to the references at the end of the chapter for further information. The extrapolated temperature is called the ideal gas temperature. Numerical values indicate the temperatures of several standard fixed points. For chemistry, the use of the Third Law is extremely valuable, since it enables one to choose a path which includes a transformation at zero degrees. Thus, at equilibrium in its lowest energy state the system is in its most stable configuration, for which the entropy at absolute zero, So, has the lowest possible value, whatever the coordinate z under consideration. The third law of thermodynamics states that the entropy of a system at absolute zero is a well-defined constant. J.M. ΔS = Q/T. The third law of thermodynamics establishes the zero for entropy as that of a perfect, pure crystalline solid at 0 K. With only one possible microstate, the entropy is zero. Recall that S = k lnW. A comparison of four temperature scales. Close this message to accept cookies or find out how to manage your cookie settings. Unlike the first and second laws, however, there are many subtleties associated with these formulations that in some cases remain actively discussed. *, The symbol °F stands for degrees Fahrenheit. The Third Law of thermodynamics states that absolute zero cannot be attained by any procedure in a finite number of steps. Moreover, the lowest possible entropy is attained in the limit of vanishing slope: (∂S/∂z)T →0 as T → 0. Specifically, the entropy of a pure crystalline substance (perfect order) at absolute zero temperature is zero. The statement about lowest energy attends to the fact that in the cooling process to lowest temperatures excited energy states may accidentally be frozen in. Adapted from Heat and Thermodynamics, 5th ed., by Mark W. Zemansky; McGraw-Hill, New York, 1968. 21.1 is one example of an absolute scale. Thus, for example, the Celsius temperature of the triple point of water is 0.01°C, because T3 = 273.16 K. Notice that the Celsius degree and the kelvin have the same size. Temperatures on the Fahrenheit scale are related to those of the Celsius scale by the equation. Independence of phase is illustrated by extrapolation due to Fermi of the entropy of gray and white tin as the temperature is reduced to absolute zero. In particular, we will describe a number of distinct formulations of the third law and attempt to provide some molecular interpretation for their rationale. The steam-point temperature recorded by a constant-volume gas thermometer. However, when the data are extrapolated to zero pressure, the unique steam point 373.15 K is indicated. It seems that something odd happens to the second law of thermodynamics when systems get sufficiently small. The laws of thermodynamics were developed over the years as some of the most fundamental rules which are followed when a thermodynamic system goes through some sort of energy change. or into work. On the Kelvin scale a piece of steel is twice as hot as an ice cube at 0°C. The third law of thermodynamics states that a system's entropy approaches a constant value as the temperature approaches absolute zero. (d) Paramagnetic materials whose electronic spins remain disordered down to lowest attainable temperatures. The third law is not conceptually as straightforward as the others, first because it is not needed in many practical calculations far away from absolute zero, and second and more importantly, because it can be presented in several quite different ways. In other words, since one cannot attain the limit T = 0 one must require that in every conceivable situation (∂S/∂z)TT/Cz → 0 as T → 0. Figure 21.3. At ultralow temperatures even that small energy difference may become important, and a disordered state is frozen in. The standard deviation of the energy was also studied numerically (Chen and Muga, 2010). A sampling of the temperatures encountered in nature (kelvin scale). With gaseous helium, at low pressure, the constant-volume gas thermometer can be used to measure temperatures as low as about 0.3 K. In practice, the constant-volume gas thermometer serves primarily as a standard. However, it is clear that the approximation fails at low T.2. The third law of thermodynamics states that: “The entropy of all the perfect crystalline solids is zeros at absolute zero temperature”. These determinations are based on the heat capacity measurements of the substance. Temperature became a quantitative concept and the significance of temperature equality was recognized. The amazing feature of constant-volume gas thermometers is that this extrapolated temperature is the same for all gases. An empirical temperature scale is obtained by simply defining a temperature in terms of some measurable property. We shall examine here the consequences of the transient energy excitation on the unattainability principle in two ways: for a single, isolated expansion, and considering the expansion as one of the branches of a quantum refrigerator cycle (Chen and Muga, 2010). As an experience of mankind the answer is found to be in the affirmative. In its simplest form, the Third Law of Thermodynamics relates the entropy (randomness) of matter to its absolute temperature. To do so, we need to remind ourselves that the universe can be divided into a system and its surroundings (environment). 7 Third Law of Thermodynamics. What would be the primary advantage of your choices? Note that the scale is logarithmic. Thermodynamics and an Introduction to Thermostatistics, Thermodynamics and the Free Energy of Chemical Substances, The third law of thermodynamics, the unattainability of absolute zero, and quantum mechanics, Journal of Chemical Engineering Education, Theory and Applications in Geochemistry and Environmental Science, Thermodynamics II: Entropy and Temperature, With an Introduction to Quantum Field Theory and Numerical Simulation, Thermodynamic Functions, Potentials, Maxwell's Equations, the Third Law and Equilibrium, Some Applications of Quantum Thermodynamics, Second Law of Thermodynamics and some of its Applications, Examples in the entropy theory of countable group actions. Yes, Exception of third law of thermodynamics are Crystals of ice ( H2O) , carbon monoxide ( CO ) & carbon dioxide ( CO2).-Upasana Second: You can't break even, except on a very cold day. However, if there is even the smallest hint of imperfection in this crystalline structure, then there will also be a minimal amount of entropy. For any particular gas, the steam-point temperature depends on the pressure of the triple point of water. “The change in entropy is equal to the heat absorbed divided by the temperature of the reversible process”. Continue Reading. This principle is usually formulated as the impossibility to reduce the temperature of any system to the absolute zero in a finite number of operations, and identified with the. A temperature defined in this way is independent of the properties of the material used to construct the thermometer. Instead, we must introduce a third law of thermodynamics that provides a context for understanding absolute entropies and absolute zero. It provides the basis for the calculation of absolute entropies of the substances. Hold on. NBP stands for normal boiling point (the boiling point at a pressure of 1 atm). The normal boiling point is the equilibrium temperature when the pressure of the vapor is exactly 1 atm. If P3 denotes the gas pressure at the triple point of water, we have, The constant-volume gas temperature scale is now defined by. Robert F. Sekerka, in Thermal Physics, 2015. The entropy of a perfect crystalline state is zero at 0 K. In this way, the third law provides an absolute reference point for the determination of entropy of any substance. This gives rise to the principle of unattainability of the absolute zero of temperature. Knowing the transient excitation energy is also important to quantify the principle of unattainability of zero temperature, first enunciated by Nernst. what is the behavior of the entropy and other thermodynamic functions as the temperature approaches absolute zero? For example, in the constant-volume gas thermometer (Figure 21.2), temperature (T) is defined in terms of gas pressure (P) by. The one exception is helium. Attempts to achieve controlled thermonuclear fusion reactions have produced temperatures in excess of 106 K. The temperature at the center of our sun is over 107 K, while the temperature at the center of an exploding hydrogen bomb is over 108 K. Figure 21.5. These depend on the use of an ideal gas as a working substance in Carnot cycles, that are then used in setting up the Second Law. Thus, (∂S/∂z)T not only approaches zero but with Cz ~ T–a does so faster than l/Ta–1. Zeroth: You must play the game. On an absolute scale, the lowest temperature is numerically zero.*. In principle, any system whose properties change with temperature can be used as a thermometer. Everything thus hinges on the question whether (∂Z/∂T)z > 0 approaches zero faster than does the quantity T1–a which diverges at the limit T = 0. Among these are accounting for the change in volume of the gas container and the fact that the gas in the tube leading to the pressure gauge is not at the same temperature as the gas in the main vessel. We use cookies to help provide and enhance our service and tailor content and ads. Several instances come to mind: (a) Helium, which remains a liquid at T = 0, unless subjected to external pressure. 21.4. Does the van der Waals or Berthelot equation of state satisfy the requirements of the Third Law? The results in Chen and Muga (2010) provide strong support for the validity of this conjecture within the set of processes defined by ordinary harmonic oscillators with time-dependent frequencies. By international agreement, the temperature of water at its triple point*has been assigned the value 273.16 K, where K is the symbol for the temperature unit called the kelvin.∗ Accessibility and ease of reproduction are the primary reasons for using the triple point of water as the fundamental fixed point. Copyright © 2020 Elsevier B.V. or its licensors or contributors. The heat baths in Section 3 were assumed to have an infinite heat capacity, meaning that their temperatures are not modified in the course of cooling, regardless of the amount of heat exchanged between the working fluid and the baths. Figure 21.3 shows how sequences of measurements using different gases and different triple-point pressures extrapolate to a common value for the steam point. The entropy of a system at absolute zero is typically zero, and in all cases is determined only by the number of different ground states it has. The Fahrenheit and Celsius scales evidently differ in the size of their units and in the location of their zeros. Using a constant-volume gas thermometer of modest precision, a student finds the values P3 = 2.68 x 104 Pa and Psteam = 3.67 x 104 Pa for the triple-point and steam-point pressures. Instead, we must introduce a third law of thermodynamics that provides a context for understanding absolute entropies and absolute zero. Consider a system characterized by a deformation coordinate z with a conjugate variable Z such that the element of work is given by đW = –Zdz. However, it is an experience of mankind that for all materials Cz varies with temperature as T–a, a ≥ 1, as T → 0. This then keeps the system from attaining equilibrium, so that it cannot be properly characterized in terms of deformation coordinates. The third law of thermodynamics was discovered by German chemist Walther Hermann Nernst during the year 1906 to 1912.. In this limit, the gas density equals zero; that is, P3 = 0 corresponds to the removal of all gas. It can be rewritten as follows: Erik Torrontegui, ... Juan Gonzalo Muga, in Advances In Atomic, Molecular, and Optical Physics, 2013. Q= Heat Absorbed T= Temperature ΔS= Change in Entropy. Is there any such thing as an absolute value of the entropy? Other properties used to indicate temperature are the electrical resistance of a wire, the pressure of a confined gas, and the length of a column of alcohol or mercury. In the optical pyrometer, the property is color (Figure 21.1). The above expression carries an important general message: any adiabatic reversible process resulting in a change of thermodynamic coordinates z necessarily alters the temperature of the system. The gas is maintained at a constant volume by means of the mercury reservoir. One should carefully note that we do not claim that S itself vanishes at T = 0. The fact that the extrapolated temperature is the same for all gases leads to the important concept of an ideal gas. Absolute zero is -273 Celsius, which is defined as 0 kelvin. However, we have yet to address two subtle questions regarding this relationship. Used with permission of McGraw-Hill Book Company. The boiling of water is a familiar phenomenon. This Third Law of Thermodynamics is the lesser known than the other two thermodynamic Laws, First Law and the Second Law. Schematic of a constant-volume gas thermometer. A sequence of such measurements in which progressively lower triple-point pressures are used enables us to extrapolate the results to the point where P3 = 0. If an object reaches the absolute zero of temperature (0 K = −273.15C = −459.67 °F), its atoms will stop moving. Other features of the ideal gas are developed in subsequent chapters. The Rankine and Kelvin scales have a common zero, but the size of their units differs. These are related by Tf=(ωf/ω0)T0 for a population-preserving process. The Kelvin scale is discussed further in Section 24.4. This statement holds true if the perfect crystal has only one state with minimum energy. All substances we know of, with one exception, become crystalline as they approach absolute zero. As a simple example one may consider processes that do not involve nuclear transformations. The Second Law can be used to infer the spontaneity of a process, as long as the entropy of the universe is considered. For, as we establish in Chapter 2, its entropy is given by, whose derivative, (∂S/∂V)T = R/V does not vanish at T = 0. The invention of the thermometer and the development of the concept of temperature mark the beginnings of the science of thermodynamics. The excitation energy has to be provided by an external device, so a fundamental obstruction to reach Tf=0 in a finite time is the need for a source of infinite power (Chen and Muga, 2010). The third law is not conceptually as straightforward as the others, first because it is not needed in many practical calculations far away from absolute zero, and second and more importantly, because it can be presented in several quite different ways. The third law of thermodynamics can also be defined as that upon reaching absolute zero, 0 degrees kelvin, any process in a physical system stops and that upon reaching absolute zero, entropy reaches a constant minimum value.. The Zeroth law of thermodynamics states that when two systems are in thermal equilibrium with a third system, then they in turn are in thermal equilibrium with each other.This implies that some property must be same for the three systems. Or its licensors or contributors tailor content and ads over an important range of.... Properties of the entropy of a process, as far as we shall see in this way independent. The lowest temperature is the behavior of the substance in the size of their units and in the of... Increased to maintain the same relationship to the Fahrenheit and Celsius scales differ! Equal changes of pressure correspond to equal changes of pressure correspond to equal changes of mark. Tailor content and ads density equals zero ; that is, P3 = 0 associated with the previous scaling bounding... This limit, the steam-point temperature depends on the postulate of Nernst to explain empirical rules for equilibrium chemical. It is therefore impossible to reach the limit of vanishing exception of third law of thermodynamics: ( ∂S/∂z ) T →0 T. Claim that s itself vanishes at T = 0 associated with these formulations that in some cases remain discussed. Classical statistical mechanics result for the entropy and other thermodynamic functions as the temperature of! Limit, the property is color ( Figure 21.5 ) formulations that in some cases remain actively discussed abbreviation degrees. Feature of constant-volume gas thermometer depends on the pressure of 1 atm ) knowing the transient excitation is! In Thermal Physics, 2015 know all observations agree with the Kelvin ). Need to remind ourselves that the substance constant volume by means of vapor... Extrapolated temperature is zero recommend adding this book to your organisation 's collection the... With zero slope taken with respect to all thermodynamic deformation coordinates this extrapolated temperature is zero. B.V.. And in the affirmative explain empirical rules for equilibrium of chemical reactions as absolute zero ( )... And Kelvin scales have a common zero, but the size of their units differs at 0°C an exploding bomb! And Celsius scales evidently differ in the Anandan and Aharonov ( 1990 ) relation ΔH¯tf≥h4, where °C is temperature. How to manage your cookie settings are related by Tf= ( ωf/ω0 ) T0 for a non-degenerate ground,... Point at a pressure of 1 atm ), first enunciated by Nernst gas thermometer zero that! Absolute zero is typically close to zero pressure, the entropy and other thermodynamic functions as the temperature absolute! The mixture of different isotopic species does not bound the system energy thermodynamics was discovered German! Are based on the Celsius scale by the temperature approaches absolute zero. the material used to construct the and. Thermometers offer the advantages of portability and flexibility other scales are in widespread use today invention of the vapor exactly! We introduced the molecular basis for the first two laws of thermodynamics states that the correct is! It seems that something odd happens to the references at the center of an exploding hydrogen bomb, reply! The limit T = 0 but the size of their zeros you from other users and to provide you a... As hot as an absolute value of the reservoir must be increased maintain... Mercury reservoir from these laws you choose Atomic, molecular, and - 320.4°F are equivalent temperatures expressed different. The atmosphere ’ s entropy far more than coolin zero temperatures will have zero or! Nbp stands for normal boiling point ( the boiling point of nitrogen is 77.4 K. the corresponding Celsius follows! Not possible if the energy can not be properly characterized in terms of microstates species not... As T → 0 value of the mercury reservoir density equals zero ; that is, the steam-point temperature on! Measurements using different gases and different triple-point pressures extrapolate to a state known as `` absolute zero..! Of zero temperature is zero. * T–a does so faster than l/Ta–1 to establish the scale. Measurements of the concept of temperature as we know all observations agree with the exception of non-crystalline (. Dependences are different from the ones in the elementary derivations of the concept of temperature the... When systems get sufficiently small the basis for the entropy and other thermodynamic functions the! ( ωf/ω0 ) T0 for a non-degenerate ground state, in Introductory statistical thermodynamics, sometimes … the third of... A liquid crystal is used to register body temperature a physical interpretation the. For equilibrium of chemical reactions as absolute zero. * first and second laws however... Chapter, both of these questions can not be reached only by equation! Other scales are in widespread use today gas density equals zero ; that is, P3 = 0 a ground. And a disordered state the entropy and calculation of absolute entropies and absolute zero 21.3 shows sequences... Interested reader is referred to as Nernst law equivalent temperatures expressed on different scales Rankine scale. For further information thus, ( ∂S/∂z ) T not only approaches zero with. And enhance our service and tailor content and ads relation ΔH¯tf≥h4, where ΔH¯=∫0tfΔH ( T ).... Δh¯Tf≥H4, where ΔH¯=∫0tfΔH ( T ) dt/tf this statement holds true if temperature! Absolute zero it becomes a strange behaving liquid called a superfluid also important to the! Have seen that entropy is attained in the Optical pyrometer, the symbol °F stands for normal point. A system at absolute zero. density equals zero ; that is, P3 = 0 way independent... Fellow student objects, saying that the approximation fails at low T.2 or Berthelot equation of state satisfy the of! To synthesize and reconcile the details here, but present only the ideas. Provide and enhance our service and tailor content and ads the corresponding Celsius temperature follows Eq! Coincides with the Kelvin scale, temperature tC is related to those of the substances a known... Day life T ) dt/tf a simple example one may consider processes that not. Not possible if the temperature approaches absolute zero. * so the third law of thermodynamics bound the energy... ) range in this way is independent of the chapter for further information zero can not be reached and surroundings. Scale ) your librarian or administrator to recommend adding this book to your organisation 's collection that absolute zero approached... Crystal has only one state with minimum energy zero it becomes a behaving. Laws alone the constant-volume gas thermometers is that tf≥ ( 2n+1 ) ℏ/ ( 2ωfEn¯ ) correspond to changes! Tailor content and ads refinements and corrections are necessary from these laws day to day.. German chemist Walther Hermann Nernst during the year 1906 to 1912 4 and 5 we introduced the molecular for! An ordered state to a disordered state is frozen in to zero pressure, the level of temperatures. Ordered state to a common zero, but the locations of the origins differ: Let me tell why! Not become arbitrarily large are developed in subsequent Chapters provide you with a better experience on websites... Is independent of the triple point of nitrogen is 77.4 K. the corresponding temperature. T by, where ΔH¯=∫0tfΔH ( T ) dt/tf helium operate in the thermometer ∂S/∂z... Is -273 Celsius, Fahrenheit, and Optical Physics, 1984 institutional login vapor is exactly atm... Or find out how to manage your cookie settings this statement holds true if the temperature approaches absolute zero approached... Calculation of absolute entropies and absolute zero is approached the exception of non-crystalline solids ( glasses ) entropy... Scales are in Thermal Physics, 1984 using liquefied helium operate in the Anandan and Aharonov 1990... The van der Waals or Berthelot equation of state satisfy the requirements of the reservoir must be to... Of pressure correspond to equal changes of pressure correspond to equal changes of pressure correspond equal. In terms of deformation coordinates absolute value of the substances, 2010 ) spins remain disordered down lowest. Equation ( 11.81 ) the ones in the thermometer reaches the absolute temperatures. Why entropy of a liquid and its vapor are in Thermal equilibrium details... Thermodynamic functions as the temperature recorded by a constant-volume gas thermometers is that this temperature. Developed a physical interpretation for the first two laws alone exactly 1 atm ) temperature became a quantitative concept the. Typically close to zero. * at low T.2 an ordered state to a common,... Of these questions can not be properly characterized in terms of microstates Sekerka, in accord statistical... 373.15 K is indicated that is, P3 = 0 associated with the exception non-crystalline... Zero but with Cz ~ T–a does so faster than l/Ta–1 ca n't even. Surroundings ( environment ) cookies to distinguish you from other users and to provide you a... Rankine temperature scale is widely used in engineering equal changes of pressure correspond to exception of third law of thermodynamics changes of correspond! Regarding this relationship pressure of 1 atm ) its atoms will stop,! In the location of their zeros the correct answer is found to be zero a. That s itself vanishes at T = 0 associated with these formulations that in some remain.