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1-2.
14.
EAS 6140 Thermodynamics of the Atmosphere and Ocean
Fall 2006 -Exam 1
Evaluate the thermal expansioncoefficient, a, for an ideal gas, where
a=-p-l
~~
(2.e
)
aT p
()1 = -~
p=~
.L
p
~
(
~L
aT
)
p
a -\
~ = -P RI I -#-
)p
= -T-
A;
3-5. The following expressionis a generalized expressionfor adiabatic processes,that is
applicable to the oceanand also to non-ideal gas planetaryatmospheres:
dT -aT
dj;:-'i>Cp
From this expression, derive an expressionfor the adiabatic lapse rate for an ideal gas.
ot'T
~
~,
~=pc;
oIT
o{:z.
~
Cp
~T -~
r
= --;ji"
-Cp
-~~'i.
-~
-
-,f
-
Cop
~?~v 0\1'0
Consider the ~ob-llric cooling of air (ideal gas, moist but unsaturated). State whether the
following properties will increase,decrease,remain the same
6. Temperature
ole~se,
7. Potentialtemperature.
8. Pressure ~11Kne..
9. Density
1Y1
CA"~,e..
10. Internalenergy aea-ec-se,
11. Enthalpy l2feu~~f.12. Entropy tAe~se.
'3. partial pressureof water vapor &o4..tY\.L.
Watervapor mixing ratio -~~
,
15. Relativehumidity. ~.~~~~-
16.
22-23.
24-25.
20-21.
Consider a dry atmosphere,ideal gas, undergoing expansionwork.
Write an expressionfor expansionwork, intensive.
r.A/A)= -potv
17. What is the internal energy for an ideal gas?
0(fA.= tJt." + ollA:>=- of0 -pot 'J = CIIotT
18-19. Write the first law of thermodynamics in internal energy form (ideal gas, expansionwork,
intensive) for an adiabatic process (use results from #16, 17)
Starting from the equation in #10-11, write the combined first and second law in internal energyform,
for a reversible process.
Starting from the equation in #22-23, derive an (integrated) expression for the.change in entropy,
[\.11,for a reversible e~pansionof an ideal gas between VI and Vz,and TI and Tzo
Cv
J.T
-:-
T
+
R
olY.1
~
=- Q\rt
Consider the following figure for # 19-22. Consider an ideal gas confined in an insulated chamber of
volume VI at a pressure PI and temperature TI. Let this gas expand into an insulated evacuated vessel
of volume V 2 until it fills both vessels. No work is done (becauseof expansion into a vacuum).
Vl
V2
Write an expressionfor the change in entropy as a result of this expansion (if you can'tfigure
out the expression, state whether entropy wil~~~~decrease,
remain the same).
26-27. The initial water vapor mixing ratio is WI' After ~E~,
(increase, decrease,remain the same)? Explain briefly.
~n-<.£-Wt\,(e,y
V'117f~~11
~'o
Mil AIItfl h'\.,( e;I(.o(,..,\'t
~,
would the mixing ratio
~
w Ii ~ ~J. ::. i
,So ~
~";i
7 a:fit'" Ml/11t1-jilYl,.-
rwt,'v wit-£,~It
W
5~
28-29, After the expansion, the relative humiHity would (increase, decrease,remain the same).
Explain briefly.
~.-:
Be ~~k~
.,. ~#~ ~
~
-..~~~_. ~S" '
!:!::!.:!:
,...,,~.fIt..-e1;"
fJ.:).,i"'i\..r'(':"
~ ~~~.~
LVs
-~
-~~
j)~R~~~"
30. Entropy will (increase, decrease,remain the ,same)in a cyclic reversible adiabatic p~ocess ..t}.
~'"
*~.
31. Entropy will (increase, decrease,remain the same) constantvolume heating
irll(l'/~
32. Entropy will (increase, decrease,remain the same)for isothermal expansion
irilWeA.sf-
33. If entropy increases,the potential temperature
c~'t-
remainsthe same, can't tell)
-U..(l (increases, decreases,
-
"
~
34. If the environmental lapse rate is 6.5vC/km, the potential temperature.. I r.,CIi
(decreases, increases, remains constant, can't tell) with height
v
35. The potentialtemperatureora parcelof air with T=288 Kat p=900hPa(mb) is
reMtI( ~
(greaterthan,lessthan,equalto) 288 K.
36-38.
Recall the definition of potential temperature
8= T(~)Pfcp
Derive an expression for potential density, Po' for dry air, which is defined as the density that
dry air would attain if it were transformed reversibly and adiabatically from its existing
conditions to a standard pressurePO. The expression should be a function only of the density,
pressure (and standard pressure), the specific heat at constant pressure and the specific heat at
constant volume.
.L
p v:.
RT
(
w. I I/~ P
r-v~
5tMe.
~~t.,:\J'Y\
l
~
p~
So we, rt
(fl.T
po~ ~9R.~
p-::fR'T
6 ~ -r ( 1.0-)R/Cp
p
~
(-.
~
':: -~~ (.f)f?/cr
1::; i!.
t
11.
r
P
r
P
yeco,M,
:::.)
::)
;::."')
i.!
-F ' rP# I' f ':: ~) ~
~
p"r , )~ w
.p
e. :: f ("i)1f
/
Consider a thermodynamic system that initially consists of liquid water in equilibrium with water vapor.
The system is heated isobarically, until all of the vapor evaporates. During the phase change from vapor
to liquid, state whether the following variables increase, decrease,or remain the same".
39. Temperature
W.Itr\D.A\'\«,
40. Specific volume
.S"~
.\rI-'tI'f~
41. Entropy
:~~
42. Enthalpy
;n.cIY~
43. Gibbs Free Energy
s~
44. partial pressure of water vapor -rn~45. Water vapor mixing ratio
;rI-U~
46. Relative humidity ~~t«~~---=47. Virtual temperature -~t:AI.~~
48. Potential temperature
=-~
s~
ConsIder that the vertical profile of specific humidity, qv, has the following form from the surface to top
of atmosphere: qy= qo(p/Po).'H;;,where .'H;;is the surface air relative humidity.
49-52. Derive an (integrated)expressionthe precipitable water over the entire vertical extent of the
atmosphere
'tv
::")
::
~
...J
-
Wv
f"
IV vi.; ::
-IV
hwt,,~
f~
ItIN"
e"~fJ.~
w
~
tf
Joo '",a,o(j
~
~ ""'(~1..,.!!:f
I ,V"
) f~
~
, ::. J!~ ~v(i) H~
1
-<t-~
p..H.
2-J
.
53-56. Derive an (integrated) expression for the optical dep't'hof water vapor over the entire vertical
extent of the atmosphere, where You can assumethat the absorption coefficient is constant with height in
the atmosphere.
"'-'/G':..-dl'
M5~
~';-'O
sec'b
:::.)~
41 ~);l we- ~,
~-r/..i) &Z~ -f\lk~t:ii~
f" -;.f J."tv
SI
;,.('J~
k)\ .II
~t
-I;, ~,(f.)H. -qdf
~~
J-~