Catholic Health Services of Long Island Phase Diagram & STP Conditions HW Need help to create a substance and identify its states of matter under standard

Catholic Health Services of Long Island Phase Diagram & STP Conditions HW Need help to create a substance and identify its states of matter under standard temperature and pressure (STP) conditions.

 Create a 4-page paper (not including the title and references pages) in a Word document for your response.

 Use APA format for the paper, title page, references page, and in-text citations.

 Develop an introduction and conclusion for your paper.

Step 1. Imagine and Name

Imagine a new substance, and give it a name. Under STP conditions, which state of matter would you like for your new substance to be?

Step 2. Create

Create a phase diagram to correlate with your STP conditions. Place temperature on the x-axis and pressure on the y-axis.

Step 3. Label

Use a blue line to separate the solid and liquid regions. Use a red line to separate the liquid and gas regions. Use a green line to separate gas and solid regions. Identify the triple point and critical point. Label the phase diagram sections.

Step 4. Explain

Explain sublimation point, condensation point, and melting point.

Step 5. Describe

Describe the phase changes that take place when the pressure is increased from STP conditions.

Step 6. Label and Interpret

Randomly place three dots in the solid, liquid and gas regions. Interpret the temperature and pressure of those dots. 999
REVIEW
Can. J. Phys. Downloaded from www.nrcresearchpress.com by DIRECTORATE OF COLDWATER FISHERIES RES on 10/13/15
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Black hole chemistry1
David Kubizňák and Robert B. Mann
Abstract: The mass of a black hole has traditionally been identified with its energy. We describe a new perspective on black hole
thermodynamics, one that identifies the mass of a black hole with chemical enthalpy, and the cosmological constant as
thermodynamic pressure. This leads to an understanding of black holes from the viewpoint of chemistry, in terms of concepts
such as Van der Waals fluids, reentrant phase transitions, and triple points. Both charged and rotating black holes exhibit novel
chemical-type phase behaviour, hitherto unseen.
PACS Nos.: 04.70.-s, 04.70.Dy, 64.60.-i.
Résumé : Traditionnellement, on identifie la masse d’un trou noir avec son énergie. Nous présentons ici une nouvelle perspective de la thermodynamique d’un trou noir, qui identifie la masse du trou noir avec son enthalpie chimique et la constante
cosmologique avec la pression thermodynamique. Ceci mène à une perception des trous noirs correspondant à un point de vue
chimique utilisant des concepts comme les fluides de van der Waals, les transitions de phase rentrantes et les points triples. Ceci
fait ressortir chez les trous noirs chargés et en rotation un comportement nouveau de type phase chimique, jamais vu auparavant. [Traduit par le Rédaction]
Our best window as to the underlying structure of quantum
gravity comes from black hole thermodynamics. By now the parallel with standard thermodynamics is well known
Energy
E ↔
Temperature T ↔
Entropy
S ↔
M
␬
2␲
A
4
Mass
Surface gravity
Horizon area
(d ⫺ 3)M ⫽ (d ⫺ 2)
in units where G = c = ប = kB = 1, where on the left-hand side we see
the basic thermodynamic quantities of a physical system, and on
the right their counterparts in black hole physics. However this
same correspondence is not quite captured by the first law of
thermodynamics, which reads
dE ⫽ TdS ⫺ PdV ⫹ work terms
↔
dM ⫽
␬
8␲
dA ⫹ ⍀d J ⫹ ⌽dQ
where the ⍀dJ and ⌽dQ terms are understood as thermodynamic
work terms. However there is no counterpart of the “pressure–
volume” term. Where might it be [1]?
Interesting new developments have recently taken place that address this question, leading to a picture in which the mass of a black
hole is interpreted as the enthalpy of space–time. This rather novel idea
originates from a consideration of the Smarr relation [2, 3]. Originally
obtained in d = 4 dimensions [4], it has the form (d – 3) M = (d –
2)TS for static asymptotically flat d-dimensional black holes, whose
metrics are
ds2 ⫽ ⫺f(r)dt2 ⫹
dr2
f(r)
where f(r) = 1 – [1/(d – 2)](16␲/␻d–2)(M/rd–3) + …, and d⍀2d is the line
element of the unit sphere Sd with a volume ␻d = 2␲(d+1)/2/⌫[(d + 1)/2].
The inclusion of a cosmological constant necessarily modifies the
Smarr relation, as can be shown from a consideration of the scaling properties of the extensive thermodynamic quantities or from
geometric arguments [1, 3]. Regarding M = M(A, ⌳), Euler’s theorem implies that
⫹ r2d⍀2d⫺2
(1)
⭸M
⭸A
A⫺2
⭸M
⭸⌳
⌳
(2)
and because T = 4(⭸M/⭸A), (2) suggests that we regard P = –⌳/8␲ =
[(d – 1)(d – 2)]/16␲ℓ2 as a thermodynamic variable [5, 6], whose
conjugate is V = –8␲(⭸M/⭸⌳). Putting this together, we have
(d ⫺ 3)M ⫽ (d ⫺ 2)TS ⫺ 2PV and dM ⫽ TdS ⫹ VdP
(3)
as the modified Smarr relation and extended first law of thermodynamics. We now have the complete thermodynamic correspondence in Table 1, where the black hole work terms are 兺i⍀idJi + ⌽dQ
for multiply rotating and charged black holes. If included, then
the more general Smarr relation
d⫺3
d⫺2
M ⫽ TS ⫹
兺 共⍀ ⫺ ⍀ 兲J ⫺ d ⫺2 2PV ⫹ dd ⫺⫺ 32⌽Q
i
i
∞
i
(4)
i
holds for all possible [d/2] rotation parameters, where the quantities
⍀∞i allow for the possibility of a rotating frame at infinity [7, 8].
The quantity P is naturally regarded as a thermodynamic pressure, an identification that follows from a cosmological perspective, because a negative cosmological constant induces a vacuum
pressure. The mass M is then understood as a gravitational version
Received 23 August 2014. Accepted 9 October 2014.
D. Kubizňák. Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, ON N2L 2Y5, Canada.
R.B. Mann. Department of Physics and Astronomy, University of Waterloo, 200 University Ave., Waterloo, ON N2L 3G1, Canada.
Corresponding author: David Kubizňák (e-mail: dkubiznak@perimeterinstitute.ca).
1This review is part of a Special Issue entitled “Theory Canada 9 Proceedings”.
Can. J. Phys. 93: 999–1002 (2015) dx.doi.org/10.1139/cjp-2014-0465
Published at www.nrcresearchpress.com/cjp on 31 October 2014.
1000
Can. J. Phys. Vol. 93, 2015
Can. J. Phys. Downloaded from www.nrcresearchpress.com by DIRECTORATE OF COLDWATER FISHERIES RES on 10/13/15
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Table 1. Thermodynamic correspondence with black hole physics.
Thermodynamics
Black hole physics
Enthalpy
Temperature
Entropy
Pressure
First law
Mass
Surface gravity
Horizon area
Cosmological constant
First law
H
T
S
P
dH = TdS + VdP + …
M
␬/2␲
A/4
–⌳/8␲
dM = (␬/8␲)dA + VdP + …
Fig. 1. Fluid behavior of charged AdS black holes when Q = 1. (a) The P–v diagram. Isotherms decrease in temperature from top to bottom. The
two upper dark curves correspond to the “ideal gas” one-phase behaviour for T > Tc, the critical isotherm T = Tc is denoted by the thick dark
line, and the lower (red) solid lines correspond to a two-phase state occurring for T < Tc for which the oscillatory part of the isotherm has to be replaced according to Maxwell’s equal area law. (b) The P–T diagram. The coexistence line of the two black hole phases terminates at a critical point characterized by the Van der Waals mean field theory critical exponents. of chemical enthalpy, namely, the total energy of a system including both its internal energy E and the energy PV required to “make room for it” by displacing its (vacuum energy) environment: M = E + PV. That is, M is the total energy required to “create a black hole and place it in a cosmological environment”. The change in perspective on the role of black hole mass and the inclusion of ⌳ as a pressure term has recently been shown to have a number of remarkable consequences for black hole thermodynamics, making their behaviour analogous to a variety of “everyday” chemical phenomena. The first indicator was the realization that charged black holes behave as Van der Waals fluids [9, 10]. The Van der Waals equation modifies the equation of state for an ideal gas to one that approximates the behaviour of real fluids [11] 冉 P⫹ a 2 v 冊 (v ⫺ b) ⫽ T ⇔ P⫽ T v⫺b ⫺ a v2 Fig. 2. Hawking–Page transition is a first-order phase transition between thermal radiation in AdS and large stable Schwarzschild– AdS black hole. It occurs when the Gibbs free energy of the black hole becomes lower than that of the thermal radiation. For any pressure P there is a first-order phase transition. (5) taking into account the nonzero size of molecules and the attraction between them, parameterized by the constants (b, a). Here, v = V/N is the specific volume of the fluid, P its pressure, and T its temperature. Critical points occur at isotherms T = Tc, where P = P(v) has an inflection point at P = Pc and v = vc, and they obey the universal relation Pcvc/kTc = 3/8 for any such fluid. A liquid–gas phase transition takes place at temperatures T < Tc, and is governed by Maxwell’s equal area law, which states that the two phases coexist when the areas above and below a line of constant pressure drawn through a P–v curve are equal. Surprisingly, charged anti-de Sitter (AdS) black holes obey the same basic relationships. For example, the d = 4 Reissner–Nordstrom–AdS metric is given by (1), with f(r) = 1 – (2M/r) + (Q2/r2) + (r2/l2). Its thermodynamic volume is given by the “Euclidean relationship”, V ⫽ 共4/3兲␲r3⫹, where r+ is the location of the event horizon. The relation T = f (r+)/4␲ along with the preceding identification of pressure and volume yield the equation of state P⫽ T v ⫺ 1 2␲v 2 ⫹ 2Q2 ␲v 4 v ⫽ 2r⫹l2P ⫽ 2 冉43V␲冊 1/3 (6) Published by NRC Research Press Can. J. Phys. Downloaded from www.nrcresearchpress.com by DIRECTORATE OF COLDWATER FISHERIES RES on 10/13/15 For personal use only. Kubizňák and Mann 1001 Fig. 3. Reentrant phase transition. (a) The Gibbs free energy G of a singly spinning black hole in d = 6, displayed for fixed pressure P 僆 (Pt, Pz) and J = 1. There is a finite jump in its global minimum indicating the presence of the zeroth-order phase transition. As T decreases, the system follows the lower vertical solid red curve, until at T = T1 it joins the upper horizontal solid red curve corresponding to small stable black holes and undergoes a first-order large–small black hole phase transition. As T continues to decrease the system follows this upper curve until T = T0, where G has a discontinuity at its global minimum. Further decreasing T, the system jumps to the uppermost vertical red line of large stable black holes. This corresponds to the “zeroth order” phase transition between small and large black holes and completes the reentrant phase transition. (b) The three-phase coexistence diagram displayed in the P–T plane. The first-order phase transition between small–large black holes is displayed by a thick solid black curve. It emerges from (Tt, Pt) and terminates at a critical point (not displayed) at (Tc, Pc). The solid red curve indicates the zeroth-order phase transition between intermediate–small black holes; the dashed black curve delineates the “no black hole” region. where the specific volume, v = 6V/N, with N ⫽ A/l2P counting the number of degrees of freedom associated with the black hole horizon, lP ⫽ 兹Għ/c3. The P–v and P–T diagrams, Fig. 1, mimic the behavior of a Van der Waals fluid for any fixed Q , with the liquid– gas phase transition replaced by the (first-order) small–large black hole phase transition, still described by the equal-area law2. Interestingly, for the critical point we find Pcvc/Tc = 3/8, precisely as in the Van der Waals case and the critical exponents are the same as those for a Van der Waals fluid [10]; a black hole that reproduces the Van der Waals equation exactly has been recently constructed [13]. This approach likewise provides a new perspective on the renowned radiation–large black hole first-order phase transition first observed by Hawking and Page for Schwarzschild–AdS black holes immersed in a bath of radiation [14]. Such a phenomenon has a dual interpretation for a boundary quantum field theory via the AdS – conformal field theory correspondence and is related to a confinement–deconfinement phase transition in the dual quark– gluon plasma. The coexistence line in the P–T diagram, Fig. 2, has no terminal point, indicating that this phase transition is present for all values of ⌳, reminiscent of a “solid–liquid” phase transition. Even more novel thermodynamics emerges for more complicated black hole space–times, possibly in higher dimensions [15]. For example a reentrant phase transition was found for singly spinning Kerr–AdS black holes in d ≥ 6 dimensions [16]. First observed in a nicotine–water mixture [17], this phenomenon is present in multicomponent fluid systems, gels, ferroelectrics, liquid crystals, and binary gases [18]. It refers to a situation in which a system can undergo a transition from one phase to another and then back to the first by continuously changing one thermodynamic variable, all others being held constant. For d = 6 black holes the global minimum of the Gibbs free energy suffers from a finite jump (Fig. 3) for a small range of pressures, P 僆 (Pt, Pz). Three separate phases of black holes emerge, which we label as (on the Fig. 4. Triple point. Various black hole phases are displayed for a doubly spinning Kerr–AdS black hole in d = 6 and the angular momenta ratio J2/J1 = 0.05 in the P–T plane. The diagram is in many ways analogous to the solid–liquid–gas phase diagram, including the existence of a triple point where three coexistence lines merge together. Note, however, that there are two critical points. That is, the small–intermediate black hole coexistence line does not extend to infinity (as in the solid–liquid case) but rather terminates, similar to the “liquid–gas” coexistence line, in a critical point. left, a form of large black hole) intermediate, (middle) small, and (on the right) large. A standard first-order phase transition separates the small and large black holes, but the intermediate and small ones are separated by a finite jump in G, indicating a zerothorder phase transition. For any given pressure in this range, the black hole will change from large to small to large (labeled inter- 2Note however that contrary to the case of the Van der Waals fluid where the equal-area law holds both in P–V and P–v variables, v = V/N, in the black hole case it holds only when expressed in terms of the P–V variables as N is no longer a constant (see, e.g., ref. 12.) Published by NRC Research Press Can. J. Phys. Downloaded from www.nrcresearchpress.com by DIRECTORATE OF COLDWATER FISHERIES RES on 10/13/15 For personal use only. 1002 mediate) again upon lowering the temperature — giving the reentrant phase transition. Similar behaviour has been observed for Born–Infeld black holes [19]. The gravitational analogue of a triple point can also be observed once two rotation parameters are included [20]. Consider a d = 6 black hole with rotation parameters J1 and J2. In the range J2/J1 僆 (0.00905, 0.0985) the phase diagram replicates the behavior of a solid–liquid–gas system, as displayed in Fig. 4. Two distinct firstorder small–intermediate and intermediate–large black hole phase transitions are possible as the temperature increases for fixed pressure in a certain range — the two transitions terminate at critical points on one end and merge at the other end to form a triple point where the three phases coexist. Contrary to the solid– liquid–gas phase diagram, the small–intermediate black hole coexistence line does not extend all the way to infinity but rather terminates at a critical point. Similar behaviour was subsequently observed for charged AdS black holes in Gauss–Bonnet gravity [21, 22]. The physics associated with the extended black hole thermodynamic phase space is only just beginning to be explored. For example, a critical point characterized by non-standard critical exponents was recently found in higher curvature gravity [23]. The compressibility –(1/V)(⭸V/⭸P)|S,J provides an indicator of the stability of the black hole, with large compressibility implying a soft equation of state and a system verging on instability [24]. V is conjectured to satisfy a reverse isoperimetric inequality [8], a kind of Penrose inequality; if proven valid, this inequality may impose restrictions on dynamical processes in AdS black hole space–times. Implications for gauge-gravity duality are also intriguing. Because neither the existence of the reentrant phase transition ⫺1/2 , these nor of the triple point depends on a variable P ⬃ ⌳ ⬃ l⫺2 P N phenomena will take place for any fixed P within the allowed range. Consequently reentrant phase transitions and triple points will have counterparts within the allowed range of N in the dual SU(N) gauge theory. Another venue for future research is to consider asymptotically de Sitter space–time (e.g., ref. 25). Acknowledgements We are grateful to N. Altamirano, B. Dolan, A. Frassino, S. Gunasekaran, D. Kastor, A. Kostouki, A. Rajagopal, Z. Sherkatghanad, F. Simovic, and J. Traschen for the interesting collaborations that led to the work described here, which was supported in part by the Natural Sciences and Engineering Research Council of Canada and by the Perimeter Institute for Theoretical Physics. Research at the Perimeter Institute is supported by the Government of Canada Can. J. Phys. Vol. 93, 2015 through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. References 1. B.P. Dolan. In Open Questions in Cosmology. Edited by G.J. Olmo. Intech, Rijeka, Croatia. 2012. [arXiv:1209.1272]. 2. M.M. Caldarelli, G. Cognola, and D. Klemm. Class. Quant. Grav. 17, 399 (2000). doi:10.1088/0264-9381/17/2/310. [arXiv:hep-th/9908022]. 3. D. Kastor, S. Ray, and J. Traschen. Class. Quant. Grav. 26, 195011 (2009). doi:10.1088/0264-9381/26/19/195011. [arXiv:0904.2765]. 4. L. Smarr. Phys. Rev. Lett. 30, 71 (1973). doi:10.1103/PhysRevLett.30.71. 5. J. Creighton, and R.B. Mann. Phys. Rev. D, 52, 4569 (1995). doi:10.1103/ PhysRevD.52.4569. 6. T. Padmanabhan. Class. Quant. Grav. 19, 5387 (2002). doi:10.1088/0264-9381/ 19/21/306. [arXiv:gr-qc/0204019]. 7. G. Gibbons, M. Perry, and C. Pope. The First law of thermodynamics for Kerr-anti-de Sitter black holes. Class. Quant. Grav. 22 (2005), 1503–1526. [hepth/0408217]. 8. M. Cvetic, G. Gibbons, D. Kubiznak, and C. Pope. Phys. Rev. D, 84, 024037 (2011). doi:10.1103/PhysRevD.84.024037. [arXiv:1012.2888]. 9. A. Chamblin, R. Emparan, C. Johnson, and R. Myers. Phys. Rev. D, 60, 064018 (1999). doi:10.1103/PhysRevD.60.064018. [hep-th/9902170]. 10. D. Kubiznak and R.B. Mann. J. High Energy Phys. 1207, 033 (2012). doi:10. 1007/JHEP07(2012)033. [arXiv:1205.0559]. 11. N. Goldenfeld. Lectures on Phase Transitions and the Renormalization Group. Westview Press, N.Y. 1992. 12. S.W. Wei and Y.X. Liu. Phys. Rev. D, 91, 044018 (2015). doi:10.1103/PhysRevD. 91.044018. [arXiv:1411.5749 [hep-th]]. 13. A. Rajagopal, D. Kubiznak, and R.B. Mann. Phys. Lett. B, 737, 277 (2014). doi:10.1016/j.physletb.2014.08.054. [arXiv:1408.1105]. 14. S. Hawking and D.N. Page. Commun. Math. Phys. 87, 577 (1983). doi:10.1007/ BF01208266. 15. N. Altamirano, D. Kubiznak, R.B. Mann, and Z. Sherkatghanad. Galaxies, 2, 89 (2014). doi:10.3390/galaxies2010089. [arXiv:1401.2586 [hep-th]]. 16. N. Altamirano, D. Kubiznak, and R. Mann. Phys. Rev. D, 88, 101502 (2013). doi:10.1103/PhysRevD.88.101502. [arXiv:1306.5756]. 17. C. Hudson. Z. Phys. Chem. 47, 113 (1904). 18. T. Narayanan and A. Kumar. Physics Reports, 249, 135 (1994). doi:10.1016/ 0370-1573(94)90015-9. 19. S. Gunasekaran, D. Kubiznak, and R.B. Mann. J. High Energy Phys. 1211, 110 (2012). doi:10.1007/JHEP11(2012)110. [arXiv:1208.6251]. 20. N. Altamirano, D. Kubiznak, R.B. Mann, and Z. Sherkatghanad. Class. Quant. Grav. 31, 042001 (2014). doi:10.1088/0264-9381/31/4/042001. [arXiv:1308.2672]. 21. S.-W. Wei and Y.-X. Liu. Phys. Rev. D, 90, 044057 (2014). doi:10.1103/PhysRevD. 90.044057. [arXiv:1402.2837]. 22. A.M. Frassino, D. Kubiznak, R.B. Mann, and F. Simovic. J. High Energy Phys. 1409, 080 (2014). doi:10.1007/JHEP09(2014)080. [arXiv:1406.7015]. 23. B.P. Dolan, A. Kostouki, D. Kubiznak, and R.B. Mann. Class. Quant. Grav. 31, 242001(2014). doi:10.1088/0264-9381/31/24/242001. [arXiv:1407.4783]. 24. B.P. Dolan. Class. Quant. Grav. 31, 035022 (2014). doi:10.1088/0264-9381/31/3/ 035022. [arXiv:1308.5403 [gr-qc]]. 25. B.P. Dolan, D. Kastor, D. Kubiznak, R.B. Mann, and J. Traschen. Phys. Rev. D, 87, 104017 (2013). doi:10.1103/PhysRevD.87.104017. [arXiv:1301.5926]. Published by NRC Research Press Copyright of Canadian Journal of Physics is the property of Canadian Science Publishing and its content may not be copied or emailed to multiple sites or posted to a listserv without the cop... 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