This document discusses molecular speeds and distributions, heat capacities, and phase diagrams. It begins by describing the Maxwell-Boltzmann distribution of molecular speeds and how temperature affects this. It then discusses how to calculate the molar heat capacity for monoatomic and diatomic ideal gases using the equipartition theorem. Finally, it introduces phase diagrams and uses examples like water, carbon dioxide, and nitrogen to illustrate different phase boundaries and triple points on pressure-temperature diagrams.
Implication of Nernst's Heat Theorem and Its application to deduce III law of thermodynamics and Determination of absolute entropies of perfectly crystalline solids using III law of thermodynamics
Implication of Nernst's Heat Theorem and Its application to deduce III law of thermodynamics and Determination of absolute entropies of perfectly crystalline solids using III law of thermodynamics
Partial gibbs free energy and gibbs duhem equationSunny Chauhan
Partial gibbs free energy and gibbs duhem equation,relation between binary solution,relation between partiaL properties,PARTIAL PROPERTIES,PARTIAL PROPERTIES IN BINARY SOLUTION,RELATIONS AMONG PARTIAL PROPERTIES,Maxwell relation,Examples
These slides are especially made to understand the postulates of quantum mechanics or chemistry better. easily simplified and at one place you will find each of relevant details about the 5 postulates. so go through it & trust me it will help you a lot if you are chemistry or a science student.
well done
Quantum yield, experimental arrangement, reasons for high and low Quantum yield, problems, photochemical reactions, kinetics of photochemical decomposition of HI, photosensitized reaction, mechanism of photosensitization,
Partial gibbs free energy and gibbs duhem equationSunny Chauhan
Partial gibbs free energy and gibbs duhem equation,relation between binary solution,relation between partiaL properties,PARTIAL PROPERTIES,PARTIAL PROPERTIES IN BINARY SOLUTION,RELATIONS AMONG PARTIAL PROPERTIES,Maxwell relation,Examples
These slides are especially made to understand the postulates of quantum mechanics or chemistry better. easily simplified and at one place you will find each of relevant details about the 5 postulates. so go through it & trust me it will help you a lot if you are chemistry or a science student.
well done
Quantum yield, experimental arrangement, reasons for high and low Quantum yield, problems, photochemical reactions, kinetics of photochemical decomposition of HI, photosensitized reaction, mechanism of photosensitization,
The revival and transformation of Europe’s largest onshore oilfield; the Pato...Albania Energy Association
Presentation: The revival and transformation of Europe’s largest onshore oilfield; the Patos-Marinza field
Leonidha Çobo, General Manager, Bankers Petroleum Albania Ltd
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Albania an important energy hub for the Southern Gas Corridor Realistic over...Albania Energy Association
Albania an important energy hub for the Southern Gas Corridor
Realistic overview of the Albanian and West Balkan
Erlet Shaqe
Co-Founder and Chairman of AEA
aea@aea-al.org
EU Gas Demand
Composition of the European gas supply
Existing and planned import pipelines to Europe
Natural gas demand on the Balkans
The Balkan Gas Hub
The Albania/Balkan region
Regional primary energy production WB
Natural Gas Role in the Albanian Energy Balance
Ionian – Adriatic Pipeline
TAP is considered important for Albania
Albania Natural Gas Forecast
Realistic potential area for developing the gas transmission and distribution in Albania
Underground Gas Storage potentials of Albania.
*Credits AEA-Albania Energy Association
Albania investments and Hydropower development 2017
The Albania Energy Association represents the Albania energy industry, covering all renewable, power, heat and fuels.
Albania Country of Opportunities .
CILESIA E AJRIT NE AMBIENTET E BRENDESHME
Studimi i mirëqënies termike nuk mund te jete i shkeputur nga ajoe e
cilësisë së ajrit në brendësi të ambienteve të rrethuara. Vëzhgimi i
sëmundjeve në dukje (sindroma të ndërtesës së sëmurë) të çuditshme në
kushte të caktuara ka çuar në lindjen e një linje tjetër studimi që i
referohet cilësisë kimike – fizike të ajrit qe, sot njihet si IAQ (Indoor Air Quality-Cilësia e Ajrit të Brendshëm). Materialet përshtatura për ndërtime, mobiljet, gazrat e përdorur në impiante, të gjithë këto prodhojnë substanca avulluese organike ( Volatile Organic Compound, VOC– Përzierje Organike Avulli) që, në përqëndrim të pakontrolluar mund të sjellin sëmundje të natyrave dhe rrezikshmërive të ndryshme.
TREGUESIT E CILËSISE SË AJRIT
PROÇESET PSIKROMETRIKE
RRL
Ngarkesa Ftohëse
CENTRALET FRIGORIFERIKE
Metodat e permasimit te tubave te ajrit
Presentation: Trans Adriatic Pipeline (TAP) – The European leg of the Southern Gas Corridor
Shkelqim Bozgo, Country Manager for Albania, Trans Adriatic Pipeline (TAP)
Presentation: Overall analysis of the onshore sector of Albania and current developments
Dritan Spahiu, Director of Policies and Development of Hydrocarbons, Ministry of Energy and Industry
How Albanian legislation facilitates the exploration and development of hydro...Albania Energy Association
Presentation: How Albanian legislation facilitates the exploration and development of hydrocarbons
Dael Dervishi, Executive Director, National Agency of Natural Resources (AKBN)
vercoming challenges in the exploration of Albania’s high potential carbonate...Albania Energy Association
Presentation: Overcoming challenges in the exploration of Albania’s high potential carbonate structures
Peter Sider, Vice President, Engineering, Operations and Business Development, Petromanas
Presentation: The role of Southern Gas Corridor for energy security of Europe
Ilkin Aslanov, Executive Assistant to SOCAR President Advisor, SOCAR, Azerbaijan
Presentation: Environmental services for the oil and gas industry
Giannis Karakolis, North Greece Director, POLYECO S.A.
Simon Geragthy, Drilling Waste Management Expert, POLYECO S.A
Presentation: Concessions and the legal framework in Energy
Etleva Kondi, Director of Concessions, Procurement and Privatisation, Ministry of Energy and Industry
Presentation: Strengthening integration on Energy Community and investment opportunities
Lorenc Gordani, Professor of EU Law and Project Director, ACERC
Presentation: The refining sector of Albania
Christophe Darbord, Chief Executive Officer, Armo Refinery
Presentation: What is the energy investment outlook in Albania for 2020?
Entela Çipa, Advisor to the Minister of Energy and Industry
Presentation: Extractive Industry Transparency Initiative - EITI in Albania
Dorina Çinari, Director, Extractive Industries Transparency Initiative (EITI) Albania
Presentation: The National Economic Council; Fostering Business-Government dialogue
Elona Varfi, COO, National Economic Council (NEC)
Keynote Presentation from the Ministry of Energy and Industry - Electricity Directorate
Agim Bregasi, Director of Policies and Development of Electricity, Ministry of Energy and Industry
Presentation: An overview of Albania’s greatest energy source - Hydropower
Aas Agnar, Director of Governmental Affairs, Devoll Hydropower, Albania
Presentation: An overview of the mining sector in Albania
Mehmet Hasalami, Policy and Development Directorate of Mines, Ministry of Energy and Industry
Transporti me litare, perfshine ato lloje te transporti ne te cilen objekti qe transportohet c’vendoset me ndihmen e nje litari. Kjo eshte konceptuar ne terheqjen e mjetit mbartes me anen e litarit, pergjate udhes se tij, ose shtriries se tij. Historikisht tipike, ky lloj transporti perfaqesonte levizjet e mjeteve lundruese, mbi lumenj ose kanale lundruese. Sot ky lloj transporti (lumenj lundruese) eshte gjeneruar ne forma te ndryshme, sic do ta shohim ne leksionet e meposhteme. Peshen kryesore ne kete lloj transporti e ze transporti me teleferik dhe ai me ashensore, (pa permendur makinat ngritese-transportuese). Transporti me terheqje rreshqanthi, pergjate tokes, eshte pak i perhapur. (Ekzistojne studime te cilat tentojne te perfshijne kete lloj transporti, te permendur ne fund, ne formen e linjave me shina, si nje forme e transportit urban ne qytetet e medha dhe me infrastrukture te dobed). Ne ne leksionin tone do te trajtojme transportin me teleferik, dhe ai me ashensore, si dy forma transporti me mundesi perhapje ne Shqiperi. Per transportin me makina peshengritese-transportuese, nuk do ti trajtojme, per specifikat e vecanta te tyre, si dhe per detyren qe ne kemi vene ne lenden tone, Transportin publik
Kerkesa per parkim
Karakteristika e problemeve te transportit
Parkimi eshte nje aspekt i rendesishem i transportit qe ndihmon (pranon) ne praktike levizjen e automjeit. Ne qendrat urbane politika e vendqendrimeve te inspirohet nga kritere dhe objektiva te ndryshme , por ne cdo rast p[ermban nje total racionale e bazuar ne njohjen e ofertes ne veprim dhe kerkeses me te gjitha kuptimet sasiore e cilesore. Vleresimi i ofertes eshte nje problem i nje vezhgimi te paster statistikore ndersa kerkesa varion ne hapsire dhe ne kohen dhe mbi gjithshka ne sjelljen e perdoruesave ne lidhjen me politiken e pergjitheshme dhe ne vecanti me vete politiken e vendqendrimeve. Ne kete kapitull do te mundohemi te ndertojme tek disa modele te vleresimit te kerkeses, ne vecanti tek modeli gravitacional , dhe do te dallohen disa parametra qe bejne matjen e kerkeses per parkim.
Po te levizjen per sherbim transporti si nje rryme fluidi (ajri/lengu), dhe infrastrukturen si nje labirinth tubash, me pike fillimi, mbarimi, si dhe interseksione te ndryshme, ku neper kete sistem tubash kalon fluidi ne fjale. Eshte e kuptueshme qe per kontrollin ose komandimin e rastit, te vendosen disa elemente konstruktive, te cilet disiplinojne kalimin e fluidit neper nje, ose disa tuba te vecante, ose te lidhur ne njefare sistemi. Keshtu me kete lloj perfytyrimi qe ne po krijojme, mund ta njehsojme levizjen kaotike neper infrastrukture, te perdoruesave te mjeteve (shofere transporti mallrash, ose pasagjeresh, ose qytetare perdorues te mjeteve private, te cilet levizin neper infrastrukturen ekzistuese, per te realizuar nje cvendosje nga nje pozicion ne nje tjeter). Ne q. se vazhdojme te krahasojme situacionin e mesiper, ne llidhje me nje fluid vleresohen parametrat cilesor te tij si:, temperaturat dhe vetit e fizike te tij, vizkoziteti, pasha specifike, shpejtesia etj. Te cilat vleresojne levizjen e fluidit ne keto tubacione. Ne nje sistem transporti levizja e perdoruesave te infrastruktures ne nje intinerar mund te vleresohet nga disa parametra ose karkateristikat e fluksit te perdoruesave, te cilat mund te grupojme 3 faktore qe influencojne direkt per kete fenomen, ose levizje. Mjeti-drejtuesi-ambjenti, te tre keto grupime jane elmente perberes dhe influencues ne cdo lloj levizje mjetesh ne te gjithe infrastrukturen.
Ne c’do projekt transporti, ne aspektin e trajtimit te skemes klasike, sic quhet teknologjia e transportit (ai i transportit te objektit nga nje vend ne nje tjeter),.kemi kete perberje ne procesin teknologjik :
Levizja e objektit (mall ose njerez) me mjetin mbartes. Ne kapitujt e mesiperm ne i kemi permendur keto lloj mjetesh te cilet percaktojne dhe llojin e transportit, konkretisht kane keto emertime:
• transporti automobilistik (mjeti eshte automobili)
• transporti detare e lumore (mjeti eshte lundra, varka)
• transporti hekurudhore (mjeti eshte vagoni i terhequr me lokomotive )
• transporti airore (mjeti eshte aeroplani, helikopteri, balona, )
• transporti me litare (mjeti eshte teleferiku, ashensori, ose c’do makine ngritese transportuese)
Nganjehere kerkohet te parashikohet nje perspektive e zhvillimit te transportit, ose te ristrukturohet ne drejtim te permiresimit te parametrave, nje sistemi transporti. Keshtu, mund dhe duhet te behen, ridimensionimet gjenerale te modeleve matematikore, ose te relacioneve matematikore te bashkangjitura keto projektit te sistemit te transportit. Sa me siper, mund te shprehim ne forme te pergjitheshme :
Tij* = ( fk1,fk2,...fkn )= f(A,B,C,...,M) ku :
T fluksi i spostimeve ndaj 2 zonave i-j dhe me karakteristikat e ketij fluksi fk1,..fkn. te cilat japin funksionet relative A,B,C,... per sistemin e transportit.
Eshte e domosdoshme te sqarojme se, ne evolucionin e kerkeses per transport, futen parametra me te dukshme, ndersa te tjerat nuk perfillen, per vete natyren diskrete, te ketyre rasteve, ky fakt e ben qe, here pas here te rishikohen vleresimet, per zgjidhjen e sistemit kzistues, (sepse faktoret ndryshojne, si koha/ kosto/ distanca e levizjes me kemb/ parkimi/ komoditeti etj.). Ne perdorimin e modelit matematik, per nje sistem transporti, praktikisht behet nje thjeshtim i realitetit, gje qe zgjidhjet optimale duhen te merren duke i permiresuar here pas here parametrat e nje sistemi transporti, ne perputhje me kushtet aktualehere.
Si duhet ta shikojme/studjojme rrealisht nje statistike ne fushen e transportitAlbania Energy Association
Elementi baze ne ndertimin e nje projekti transporti eshte analiza e kerkes-ofert per transport. Kjo shpesh vjen nga verejtjet, ose pamjat direkte ne terren, nga menyra se si sillen ose levizin njerezit apo mallrat. Ashtu si dhe ne vleresimet e tjera per terrenin, edhe ketu kemi nje logjike te perafert, e dalluar, si me poshte:
1) Niveli i sherbimit mbi infrastrukturen ekzistuese, kjo ka te beje me grumbullimin e te dhenave rreale, statistikore mbi infrastrukturen ekzistuese, si dhe kerkesen per transport, apo sherbimet e transportit, si dhe pyetjet ose ceshtjet te tjera, qe lindin ne kete bashkeveprim kerkese-oferte
2) Modelimi simulues i nje sistemi, ne te cilen jane parashikuar vleresimet tekniko-ekonomike, nder te tjerat dhe demografike, social-ekonomike, urbanistike, etj. Si kane ndikuar ose vijuar keto elemente, gjate shfrytezimit ?
3) Treguesit Financiare, si kane qene dhe si eshte bilanci financiare i ketij sherbimi, dinamika e tij, si dhe ndonje alternative e tij, ne pasqyren e kostove e te rendimentit
Per ndertimin e nje projekt transporti, duhet detyrimisht te merren ne konsiderate shume elemente, ose kushte, te cilat konvertohen ne nje bashkesi ndervaresie e quajtur Modeli i sistemit te transportit, ky model eshte i afte ti simuloje keto kushte apo elemente, te projektit duke ne dhene rezultatet respektive. Sa me siper, nje projekt per nje sistem transporti, duhet te ece neper keto faza logjike:
Individualizimi i objeketve qe kerkojne levizje (kerkesa per transport)
Analiza sasiore e sistemit te transportit aktual (ose per disa sisteme)
Ndertimi i nje model varesie, qe perfaqeson nje sitem transporti, i cili eshte ne gjendje te funksionoje me te dhenat (dati) aktuale, si edhe me parametrat e vlerave te prespektives, ne lidhje me kete sistem qe po projektojme, se bashkeu me sistemet e tjere konkurues
Pergjithesime
Nje nder problemet qe e veshtiresojne shperndarjen, shtririen e transportit te mallrave mbi rruge, eshte koha e madhe jo levizese e mallit, ne pikat e nderthuries se levizjeve (grumbullimin, shperndarjen, vazhdim levizjeje te mallit ne destinacion) e cila presupozon kohen e perpunimit te mallit. Si dhe siperfaqet e disponueshme per kete kohe ne keto pika te nderthurjes se levizjes e te perpunimit te mallrave. Keshtu mga viti 1930-40 filloj nje koncep i ri “unifikimi i njesive te ngarkesave” kjo do te thote qe menyra te ambalazhimit te jete e paraqykyar qe ne projektim si nje element ne teknologjine e organizimit te transportit te objektit, kjo gje solli qe:
Elementi baze ne ndertimin e nje projekti transporti eshte hapesira fizike e shtrirejes se aktivitetit te sistemit te transportit, kjo per faktin e kalimeve ose te ndersjelles reciproke, te lelementeve teknike e politike te ketyre siperfaqeve. Per kete te kujtojme nje fakt te thjeshte dhe shume domethenes, ne shtrirjen e linjave nderkokmbetare te hekurudhes, dikur jo te gjitha shtetet, kishin standartin e distancave te shinave te barabarte, pra nuk rrealizohej teknikisht kalimi i vagonit te trenit nga nje shtet tek tjeti, (pasi nuk i kishin te njejta distancen e shinave), sot ky element nuk egziston (eshte eleminiur me kohe), por ekziston ndryshimi i tensionit elektrik, ne rrjetin e fuqise, per linjat hekurudhore te shteteve te ndryshme. (sot ky problem eleminohet duke ndrushuar lokomotiven, pra artificialisht krijohet ndalesa). Ne nje sistem transporti urban, siperfaqet ose zonat e hapesirave, kane rendesi te shikohen ne aspektin cilesor te tyre, nder te cilat mund te permendim, gjendja social-ekonomike e siperfaqes, prezenca e qendrave te medha tregtare e kulturore, destinacioni i aktivitetit te siperfaqes (aktiviteti kryesore i saj). etj. Por hapsira e territorit te referimit, mund te jete dytesore, ne kuptimin e rendesise se nivelit te transportit (pra nuk merret ne konsiderate karakteristikat e zones ne te cilen operohet). Hapsirat i dallojme nga elementi i pare i tyre, ndarjet poloitike.
Problemet e transportit jane bere me te mprehura dhe me kritike se asnjehere tjeter, si ne vendet e industrializuara ashtu edhe ne ato ne zhvillim. Mungesa e lendes djegese (perkohesish) nuk jane probleme qe e preokupojne sot transportin, por eshte rritja e pergjithshme e trafikut dhe e kerkesave per transport, te cilat kane si pasoje efektet e mbingarkit te trafikut, vonesat, aksidentet, si dhe problemet per mjedisin, keto te fundit kane shkuar aq laarg, sa tashme nuk mund te jene te pranueshme. Shprehemi qe keto probleme, nuk mund te kufizohen tek rruget, ose infrastruktura dhe trafiku i autoveturave.
How to Split Bills in the Odoo 17 POS ModuleCeline George
Bills have a main role in point of sale procedure. It will help to track sales, handling payments and giving receipts to customers. Bill splitting also has an important role in POS. For example, If some friends come together for dinner and if they want to divide the bill then it is possible by POS bill splitting. This slide will show how to split bills in odoo 17 POS.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
How to Create Map Views in the Odoo 17 ERPCeline George
The map views are useful for providing a geographical representation of data. They allow users to visualize and analyze the data in a more intuitive manner.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Students, digital devices and success - Andreas Schleicher - 27 May 2024..pptxEduSkills OECD
Andreas Schleicher presents at the OECD webinar ‘Digital devices in schools: detrimental distraction or secret to success?’ on 27 May 2024. The presentation was based on findings from PISA 2022 results and the webinar helped launch the PISA in Focus ‘Managing screen time: How to protect and equip students against distraction’ https://www.oecd-ilibrary.org/education/managing-screen-time_7c225af4-en and the OECD Education Policy Perspective ‘Students, digital devices and success’ can be found here - https://oe.cd/il/5yV
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
For more information, visit-www.vavaclasses.com
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
2. Molecular speeds
Not all molecules have the same speed.
If we have N molecules, the number of molecules with
speeds between v and v + dv is:
dN = Nf (v )dv
f (v ) = distribution function
f (v )dv = probability of finding a molecule with speed
between v and v + dv
4. Distribution = probability density
f (v )dv
v2
∫v
1
= probability of finding a molecule with speed between v and v + dv
f (v )dv = probability of finding molecule with speeds between v1 and v2
= area under the curve
Normalization:
∫
∞
0
f (v )dv = 1
5. Most probable speed, average speed,
rms speed
2kT
m
v mp =
Most probable speed
(where f(v) is maximum)
∞
∫ vf (v )dv =
=
∫
∫ f (v )dv
v
0
∞
vf (v )dv = ( ... ) =
∞
0
0
v
2
3kT
= ∫ v f (v )dv = ( ... ) =
0
m
vrms =
∞
2
v
2
3kT
=
m
8kT
πm
Average speed
Average squared speed
rms speed
6. Molar heat capacity
How much heat is needed to change by ΔT the
temperature of n moles of a certain substance?
m = nM
n = number of moles
M = mass of one mole (molar mass)
Q = mc ∆T = nMc ∆T
Q = nC ∆T
C = Mc
C = molar heat capacity
7. Constant volume or constant pressure?
The tables of data for specific heats (or molar capacities) come
from some experiment.
For gases, the system is usually
kept at constant volume
CV
For liquids and solids, the
system is usually kept at
constant pressure (1 atm)
CP
You can define both for any system.
You need to know what’s in the table!
8. Heat capacity for a monoatomic ideal gas
Average total kinetic energy K total =
3
3
NkT = nRT
2
2
3
d K total = nRdT
2
From the macroscopic point of view, this is
the heat entering or leaving the system:
3
nRdT = nCV dT
2
3
CV = R
2
dQ = nCV dT
Molar heat capacity at
constant volume for
monoatomic ideal gas
Point-like particles
9. Beyond the monoatomic ideal gas
Until now, this microscopic model is only valid for monoatomic
molecules.
Monoatomic molecules (points) have 3 degrees of freedom
(translational)
Diatomic molecules (points) have 5 degrees of freedom: 3
translational + 2 rotational)
Principle of equipartition of energy: each velocity
component (radial or angular) has, on average,
associated energy of ½kT
The equipartition principle is very general.
10. Diatomic ideal gas
K = K tr + Krot
3
2
= kT + kT
2
2
5
Average energy
= kT
per molecule
2
Average total energy
K total =
The same temperature involves more
energy per molecule for a diatomic
gas than for a monoatomic gas..
5
5
NkT = nRT
2
2
dQ = nCV dT
DEMO:
Mono and diatomic
“molecules”
5
CV = R
2
Including rotation
Molar heat capacity at
constant volume for
diatomic ideal gas
11. Monoatomic solid
Simple model of a solid crystal: atoms held together by springs.
K1 =
3
kT
2
Vibrations in 3 directions
But we also have potential energy!
PE1 = K1
For N atoms:
Molar heat capacity at
constant volume for
monoatomic solid
UN = N KE1 + N PE1
= 3NkT
= 3nRT
(for any harmonic oscillator)
d U = 3nRdT
dQ = nCV dT
CV = 3R
12. pT diagram
Melting curve (solid/liquid transition)
solid
Triple
point
Sublimation curve
(gas/solid transition)
liquid
Critical
point
gas
Vapor pressure curve
(gas/liquid transition)
13. pT diagram for water
DEMO:
Boiling water
with ice
p
solid
Critical point
liquid
demo
1 atm
gas
0°C
100°C
T
14. In-class example: pT diagram for CO2
Which of the following states is NOT possible for CO 2 at 100 atm?
A. Liquid
B. Boiling
liquid
C. Melting
solid
D. Solid
E. All of the
above are
possible.
15.
16. Melting (at ~ -50°C)
solid
liquid
Boiling (at ~ -5°C)
For a boiling transition,
pressure must be lower:.
17. Triple point for CO2 has a pressure > 1 atm.
At normal atmospheric pressure (1 atm), CO 2 can only be solid or gas.
Sublimation at T = -78.51°C
18. pT diagram for N2
At 1 atm, Tboiling = 77 K Tmelting = 63 K
Triple point for N2: p = 0.011 atm, T = 63 K
p
1 atm
solid
liquid
Triple
point
DEMO:
N2 snow
63 K
demo
gas
77 K
T
19. pV diagrams
Convenient tool to represent states and transitions from one
state to another.
Expansion at constant
pressure
p
states
A
B
process
(isobaric process)
VB
VA
V
Example: helium in balloon expanding in the room and warming up
If we treat the helium in the
balloon is an ideal gas, we can
predict T for each state:
TA/B =
pVA/B
nR
DEMO:
Helium balloon
20. ACT: Constant volume
This pV diagram can describe:
p
A. A tightly closed container cooling
down.
pB
B. A pump slowly creates a vacuum
inside a closed container.
C. Either of the two processes.
pA
B
A
(isochoric process)
In either case, volume is constant and
pressure is decreasing.
In case A, becauseT decreases.
In case B, because n decreases.
V
21. Isothermal curves
For an ideal gas, p =
nRT
V
T1 <T2 <T3 <T4
(For constant n, a hyperbola for each T )
Each point in a pV
diagram is a possible
state (p, V, T )
Isothermal curve = all states with the same T
22. ACT: Free expansion
A container is divided in two by a thin wall. One side contains an ideal gas, the
other has vacuum. The thin wall is punctured and disintegrates. Which of the
following is the correct pV diagram for this process?
A
1
Initial state
B
2
Initial state
Final state
Final state
3
C
Initial state
Final state
4
D
Initial state
Final state
23. Final state has larger V, lower p
During the rapid expansion, the gas does NOT uniformly fillV at a
uniform p
⇒ hence it is not in a thermal state.
⇒ hence no “states” during process
⇒ hence this process is not represented by line
A
1
Initial state
B
2
Initial state
Final state
Final state
3
C
Initial state
Final state
4
D
Initial state
Final state
24. Beyond the ideal gas
When a real gas is compressed, it eventually becomes a liquid…
Decrease volume at constant
temperatureT2:
• At point “a”, vapor begins to condense
into liquid.
• Between a and b: Pressure and T
remain constant as volume decreases,
more of vapor converted into liquid.
• At point “b”, all is liquid. A further
decrease in volume will required large
increase in p.
25. The critical temperature
Critical point for water: 647K and 218 atm
For T >> Tc, ideal gas.
p
Supercritical fluid
solid
Triple
point
liquid
Critical point
gas
T
critical temperature = highest temperature where a
phase transition happens.