# EAS 6140 Thermodynamics of the Earth System

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```					                   EAS 6140 Thermodynamics of the Earth System
Fall 2007 – Practice Exam 1

Units (you aren’t expected to have memorized these, but you should easily be able to
figure them out based upon dimensional analysis of equations that you have memorized
or are on the sheet of equations.

1. Specifc gas constant
J/(kg K)

2. Specific heat capacity
J/(kg K)

3. Scale height of the homogeneous atmosphere
meters

4. Gibbs energy
Joules or for specific Gibbs energy J/kg

5. optical depth__________________
unitless

Consider a hypothetical planet named Argonia, which has exactly the same physical
characteristics as planet Earth (e.g. size, distance from sun, mass of atmosphere), except
that the sole constituent of Argonia's atmosphere is Argon (Ar, molecular weight= 40).
Describe how the following thermodynamic characteristics of the Argonian atmosphere
would differ from the Earth's atmosphere (e.g. larger, smaller, the same). Briefly

* All the parameters below are related to the specific gas contstant.
6. specific gas constant (note: molecular weight of gases in the Earth’s atmosphere is
29)
Specific gas constant is defined as, Rd = R*/M
Since the molecular weight of air in Earth’s atmosphere is primarily nitrogen; N2 =
28g/mol and on Argonia it is Argon; Ar=40g/mol, this will cause the specific gas
constant on Argonia to be smaller than the specific gas const of air on Earth.

Dry adiabatic lapse rate is defined as, Γ = g/cp and cp = 5/2R
Dry adiabatic lapse rate should become more negative (i.e. Temperature should
(decrease with height more rapidly on Argonia than it does on Earth)

8. scale height (of the homogeneous atmosphere)
Scale height is defined as, H= RdTo / g ; smaller
Consider a hydrostatic atmosphere (ideal gas) with constant density.

9. Write an expression for hydrostatic equilibrium (differential form)
∂p / ∂z = - ρg

10. Write the mathematical definition of lapse rate.
T    g
             34.1 C / km  avg  6.5 C / km         (Environmental)
z   Rd

11. Derive an expression for the lapse rate of the constant density atmosphere

∂p = - ρg ∂z                  Recall Ideal gas law p = ρ Rd T
And use ρ Rd as constants

ρ Rd ∂T = - ρg ∂z
∂T/∂z = - g / Rd

12-14. Define a variable x=cp/cv. Is the following statement true or false?

1–x =– R
x     cp

 cp    cv c p       
1 
c     
  c              
 v     v cv           cv  c p
cp        c             c
     p
      p
 c           
cv            v       
Recall, R = cp– cv

cv  c p       R
               Statement is true!
cp           cp

Consider the (reversible) adiabatic expansion of an ideal gas (moist but unsaturated).
State whether the following properties will increase, decrease, remain the same

15. Temperature______decreases___________

16. Potential temperature___same______________

17. Pressure_____decreases____________
18. Density____decreases_____________

19. Internal energy____decreases_____________

20. Enthalpy________decreases_________

21. Entropy__same_______________

23. partial pressure of water vapor ______decreases_______

24. Water vapor mixing ratio ___constant__________

25. Relative humidity _____increases_________
Consider a dry atmosphere, ideal gas, undergoing expansion work.

26. Write an expression for expansion work, intensive.

dw   pdv

27. What is the enthalpy for an ideal gas?

dh = du + d(pv)

28-29. Write the first law of thermodynamics in enthalpy form (ideal gas, expansion
work, intensive) for an adiabatic process.

cpdT = dq + vdp
dq = 0
cpdT = vdp

30-31. Write the combined first and second law in internal energy form, for a reversible process.

du = Tdη – pdv

32-33. Derive an (integrated) expression for the change in entropy, , for a reversible
expansion of an ideal gas between p1 and p2, and T1 and T2.

dT    dp
d  c p      R
T      p
Rc
T2       p2        p
34-36. If T and p are related by          =
p1            what is the change in entropy, ? (Hint:
T1
use the results from the preceding question)
T         p 
From eqn 2.26b   c p ln  2   R ln  1 
T         p 
 1         2
R / cp
p                     p 
 c p ln  2 
p               R ln  1 
p 
 1                     2

Or this relation could be alternatively expressed for temperature.

In the following questions on boiling point, you may want to refer to Appendix D
(attached). Also, you may wish to refer to the critical point data for water: 647K and
218.8 atm.

39-40. The surface pressure on Planet X is 10 mb. What is the boiling point temperature
on Planet X? Explain.
Since we know that the boiling point is reached when saturation vapor pressure equals
atmospheric pressure, the new boiling point temperature is 7 oC as shown in Appendix D.

40-41. What is the change in pressure necessary to increase the boiling point of water from 99.5
to 100.5oC (refer to Appendix D)
38mb. This value came from doing the calculation not from Appendix D.

42-43. Consider the boiling temperature at the bottom of the ocean that is 5 km deep.
Pressure in the ocean increases by approximately 1 atm for each 10 m. Which of the
following points is true about the boiling point at the bottom of the ocean? Explain
a) the boiling point will be the same as at sea level
b) the boiling point will be lower than at sea level
c) the boiling point will be lower than at sea level
d) the water won’t boil at all
The vapor pressure curve has a natural upper limit at the critical point (T=647 an
p=218.8atm) beyond which the liquid phase is no longer distinguishable from the vapor
phase. Beyond this critical point water will not boil.

44-46. On an early spring day, the lapse rate above Denver (altitude 5300 feet) and its
surroundings is about the average value, two-thirds the dry adiabatic lapse rate. Air over
the 14,000 foot mountains to the west begins to flow down the slope into Denver. Do
you expect the air temperature in Denver to increase, decrease, or remain the same?
Explain.

As the wind moves up the mountains to a lower pressure (and height), it should cool. As
the air descends the leeward side of the mountain range it should warm via adiabatic
compression. This means we should expect the air temperature at Denver to increase.
47-49. Find an (integrated) expression for the optical depth of water vapor over the entire
vertical extent of the atmosphere having constant mixing ratio (~ specific humidity). You
can assume that the absorption coefficient remains constant with height.

d  k v dz
v
Recall that the mixing ratio is, wv 
d
Where,  d ( z )   0 exp  z / H 
d  k d wv dz

d  k d ( z)wv dz  kwv  d  exp( z / H )dz
  kwv  0 H exp(-z/H)

50-53. Develop a relationship between the precipitable water through the entire vertical extent of
the atmosphere and the sea surface temperature, T0. Assume:
a) the vertical profile of specific humidity, qv, has the following form: qv = q0(p/p0)H0, where
H0 is the surface air relative humidity;
b) the saturated vapor pressure can be approximated by the following expression es ~ b exp [a
(Ts -T0)]; and
c) the specific humidity can be approximated by the water vapor mixing ratio.

1
g
Wv       q v dp

Re call, q v  q 0  p  H 0
 p 
   0
0
q H
Wv  0 0
gp0              pdp
p0
2
q H p   q H p
 0 0 0  0 0 0
2 gp0   2g
Re call, q v  wv and
H 0 es
wv  H 0 ws 
p0  es
Re call, p 0  es
H 
2

Wv  0 es
2g
H 
2

Wv  0 b expaTs  T0 
2g

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