ESCI 341 Atmosherc Thermodynamcs Lesson 6 Thermodynamc Processes Reerences: An Introducton to Atmosherc Thermodynamcs, Tsons Introducton to Theoretcal Meteorology, Hess Physcal Chemstry (4 th edton), Lene Thermodynamcs and an Introducton to Thermostatstcs, Callen ISOTHERMAL PROCESSES I a rocess s sothermal (dt = 0) then or an deal gas the rst law becomes dq = d For an deal gas we can substtute or rom the deal gas law to get whch ntegrates to d dq = R T q = R T ln We can also use the enthaly orm o the deal gas law, whch or an sothermal rocess becomes dq = d When ths s ntegrated we get q = R T ln ISOCHORIC PROCESSES I a rocess s sochorc (constant olume) then the rst law or an deal gas becomes dq = c dt Ths can be ntegrated to get (assumng c s constant) ( T T ) q = c ISOBARIC PROCESSES For an sobarc rocess, d = 0 Thereore the rst law or an deal gas becomes whch ntegrates to dq = c dt
( T T ) q = c ADIABATIC PROCESSES An adabatc rocess s one n whch there s no heat transer (dq = 0) The two orms o the rst law o thermodynamcs or an adabatc rocess n an deal gas are c dt = d c dt = d I we start wth the rst orm o the rst law or an deal gas (the one nolng c ) and substtute or ressure rom the deal gas law, we get ο Integratng ths ges whch can also be wrtten as dt d c + R = 0 T c lnt + R ln R c T (1) ο We e reously shown that c c = R Thereore, we can wrte Eqn (1) as ( c c ) T and denng the rato c c γ we get c 1 T γ (2) ο Usng the deal gas law, ths equaton can also be wrtten as γ (3) or T (4) Equatons (2), (3), and (4) are known as the Posson relatons (note that the constant on the rght-hand-sde s not necessarly the same n each equaton T T γ 1 γ Posson relatons 2
The Posson relatons relate T,, and n deal gases undergong quas-statc, adabatc rocesses I you know the ntal alues o two o these arables, and one o ther nal alues, you can nd the other two nal alues by usng these relatons It s mortant to realze that Posson s relatons are only ald or deal gases undergong quas-statc adabatc rocesses! It s narorate to use them or nonadabatc rocesses POTENTIAL TEMPERATURE Potental temerature (denoted as θ ) s dened as the temerature an ar arcel would hae t were moed dry-adabatcally to a reerence ressure, 0, o 1000 mb From the Posson relaton or T and [Eqn (4)] we get (see Exercse 11) R d c 0 θ = T I an ar arcel undergoes an adabatc rocess ts otental temerature s consered WORK IN AN ADIABATIC PROCESS For an adabatc rocess the change n nternal energy s solely due to work done on or by the system, du = dw ο Note that ths s true or any system (not just deal gasses) and regardless o whether the adabatc rocess s quas-statc or not For an deal gas, du = c dt = dw ADIABATIC FREE EXPANSION I an deal gas s allowed to adabatcally reely exand, unoosed, ts temerature wll not change To see why, recall that or an deal gas undergong adabatc exanson cdt = dw In a ree exanson there s no work done, so there s no change n temerature But what about the exresson 3
dw = d? The gas exanded, so secc olume changed, so shouldn t there be work accomlshed? KEY POINT: Remember that the exresson dw = d only ales to quasstatc rocesses A ree exanson s not quas-statc, so we can t calculate work usng ths exresson EXERCISES 1 Show that or an sothermal rocess or an deal gas q = R T ln 2 For an sothermal rocess or an deal gas, show that the work done by the system s w = R T ln or w = R T ln 3 a For an sobarc rocess show that ( T T ) + ( ) u = c b Is ths true or all gasses, or only deal gasses? 4 Startng wth c dt = dq +d, dere the Posson relaton T 5 A 15-kg arcel o dry ar s at a temerature o 15 C and a ressure o 1013 mb 4
a How many moles o ar are n the arcel? (The molecular weght o ar s 2896 g/mol) b What s the olume o the arcel? c What s the secc olume o the arcel? d I 50 KJ o heat are added to the arcel whle ts olume s held constant, what s the new temerature o the arcel? (The secc heat o ar at constant olume s 717 J-kg 1 -K 1 ) 6 An arcel o dry ar s at a temerature o 15 C and a ressure o 1013 mb Heat s added to the arcel to cause t to exand It exands at constant ressure to 15 tmes ts orgnal olume a What s the new temerature o the arcel? b How much work (er unt mass) was done by the arcel durng ths exanson? c What was the change n secc nternal energy o the ar arcel? d What was the amount o heat er unt mass that was added to the ar arcel? 7 An ar arcel s at a temerature o 15 C and a ressure o 1013 mb Heat s added to the arcel to cause t to exand It exands at constant temerature untl ts olume s 15 tme t orgnal olume a What s the new ressure o the ar arcel? 5
b How much heat er unt mass was added to the ar arcel? c How much work er unt mass was done n exandng the ar arcel? d What was the change n secc nternal energy o the ar arcel? 8 A dry ar arcel at an ntal temerature o 20 C and a ressure o 950 mb s orced to rse adabatcally u a mountan sloe The to o the mountan s at a ressure o 720 mb a What s the temerature o the ar arcel when t reaches the to o the mountan? b What s the work done by the ar arcel? 9 A cylnder lled wth helum (a monatomc deal gas) has a olume o 18x10 6 cm 3, a ressure o 12x10 5 mb, and a temerature o 300K The cylnder s contaned n an eacuated room wth a olume o 16 m 3 The cylnder rutures and helum lls the room a What s the ressure n the room ater the cylnder rutures? b What s the temerature n the room ater the cylnder rutures? c What s the work done by the exandng helum? 10 A arcel o dry ar s ntally at a ressure o 900 mb and a temerature o 15 C It rses to the 400 mb leel a What amount o heat (er mass) must be exchanged wth ts surroundngs the temerature s to reman constant at 15 C durng the ascent? Wll the heat be ganed or lost by the arcel? 6
b I the arcel rst ascends adabatcally to 400 mb, and then heat s added to t to rase ts temerature back to 15 C, how much heat must be added? Is ths the same amount o heat as the reous queston? I not, why not? 11 Usng the Posson relaton show that T R ( ) d c θ = T 0 12 For a non-deal gas, wll an adabatc ree exanson result n a temerature change? Exlan 7