EAS 3603/6140: Thermodynamics of Atmospheres and Oceans
Worksheet 5 – First Law of Thermodynamics
1. The first law of thermodynamics is a statement of the conservation of
__Energy________
When an increment of heat dQ is added to a system, the energy may be used either to increase the
speed of the molecules (i.e., to increase the temperature of the system), to create motion internal
to each molecule (e.g., rotation and vibration), or to overcome the forces of attraction between the
molecules (e.g., change of state from liquid to vapor), all of which contribute to the internal
energy of the system. The internal energy of a system can increase when heat enters into the
system from the surroundings, and/or when work is done on the system by the surroundings.
From the statement of conservation of energy, we have (intensive form, using sign convention in
book)
du = dq + dw
From the law of conservation of energy, the total energy of the system plus its environment must
=
syst + U env
be constant. 0 U
2. What happens to internal energy in a cyclic process? (increase, decrease, remain the same)
remains the same
3. Is internal energy an exact differential? YES NO
Yes
4. It is convenient to define a new function called the enthalpy, h, by h = u + pv
Expand the expression for h into a differential dh = .
dh = du + d(pv)
5. Substitute for du the expression for the first law of thermodynamics in #1.
dh = dq + vdp
6. Are the following equations both statements of 1st law of thermodynamics YES NO
du = dq - pdv
dh = dq + vdp
Yes
7. Circle all of the following that are exact differentials:
a) heat
b) work
c) internal energy
d) enthalpy
Explain why you think enthalpy is or is not an exact differential.
Because all the functions used to calculate enthalpy are state functions
8. Write an expression for the internal energy of an ideal gas
du = cvdT
9. Write an expression for the enthalpy of an ideal gas
dh = cpdT
10. In a constant-pressure process, some of the added heat must be expended in doing work on
the surroundings, while in a constant-volume process, all of the heat is devoted to raising the
temperature of the substance. For the same material, how do the values of cp, cv compare?
a) cp > cv
b) cp < cv
c) cp = cv
11. From the definition of enthalpy, h=u+pv, the differential form of enthalpy can be
written as dh=du+d(pv). For an ideal gas, substitute values of dh, du, and d(pv) into this
equation to derive a value for cp - cv for an ideal gas. NOTE: this expression is
applicable only for an ideal gas. For non-ideal gases, liquids, and solids, values of cp, cv
vary with temperature and pressure.
dh  du  d ( pv)
 c p dT  cv dT  pdv  vdp  cv dT  RdT
 c p  cv  R
12. If the value of cv for a monatomic gas is cv=(3/2)R, what is the value of cp?
c p  cv  R  c p 
3
5
R  R  cp  R
2
2
The following equations are both statements of 1st law of thermodynamics
du = dq - pdv
dh = dq + vdp
For ideal gases, u and h have been shown experimentally to be functions only of T so that
du = cv dT
dh = cp dT
12. Write the 1st law in both internal energy and enthalpy forms for an ideal gas,
combining the above expressions.
cvdT = dq - pdv
cpdT = dq + vdp
Consider an ideal gas that is subject to expansion work. In the following, write
expressions of the first law (starting from equations in #12), differential form with
intensive variables, that are applicable ONLY to an ideal gas with expansion work
13. Write an expression for the first law of thermodynamics for constant volume process
(note: do not write a general expression for the first law, but one that is applicable
ONLY to constant volume processes). Is it more sensible to write this expression in
terms of the internal energy or the enthalpy?
du = dq – pdv dv=0
du = dq
internal energy
14. Write an expression for the first law of thermodynamics for isobaric process (note:
do not write a general expression for the first law, but one that is applicable ONLY to
isobaric processes). Is it more sensible to write this expression in terms of the internal
energy or the enthalpy?
dh = dq + vdp dp=0
dh = dq
enthalpy
15. Write an expression for the first law of thermodynamics for an isothermal process
(note: do not write a general expression for the first law, but one that is applicable
ONLY to isothermal processes). Is it more sensible to write this expression in terms of
the internal energy or the enthalpy?
cvdT = dq – pdv
cvdT = 0
dq = pdv
either would work
16. Write an expression for the first law of thermodynamics, enthalpy form, for an
adiabatic process. (note: do not write a general expression for the first law, but one that
is applicable ONLY to adiabatic processes)
cpdT = dq + vdp
dq = 0
cpdT = vdp
17. Starting from the expression you obtained in #16, substitute the ideal gas law for
specific volume. You should now have and equation that contains only T, p as
thermodynamic state variables.
cpdT = RT/p dp
18. Integrate the expression you obtained in #17 from (T1, p1) to (T2, p2).
T2 dT
T
R p2 dp
R p
T1 T  c p p1 p  ln T12  c p ln p12
19. Take anti-logs of the expression you obtained in #18.
R
T1  p2  c p
 
T2  p1 
By repeating #16-#19 but starting from the internal energy form of the first law, the
following expression is derived
T2
v
= v1
T1
2
Rc
v
Substitituing from the ideal gas law and expression from #7, can also write relation
between p, v
p2
v
= v1
p1
2
cp
cv
These expressions can also be written as
Tvn-1 =const
pvn =const
Tp(1-n)/n =const
where n is the polytropic index which for an ideal gas is n=cp/cv.
For more general processes do not involve ideal gas and may not be adiabatic, the
polytropic expressions are frequently used, but with an empirically-derived value of n.
Dry Adiabatic Lapse Rate:
20. Write the first law of thermodynamics, enthalpy form, adiabatic, intensive, for an
ideal gas (see #17).
cpdT = vdp
21. Starting from #20, substitute the hydrostatic equation for the appropriate part of the
expression in #20.
dp   g  dz
c p dT  v  gdz   gdz (v   1)
22. Starting from #21, write an expression for the dry adiabatic lapse rate, d (note, if
you did this correctly, you should obtain equation (2.68).)
dT v g g
d 


(v  1)
dZ
cp
cp
23. What is the magnitude of the dry adiabatic lapse rate for the atmosphere?
9.8C / km
24. How does the magnitude of the dry adiabatic lapse rate compare with:
a) the observed tropospheric lapse rate
larger than the observed one
b) the lapse rate of the homogeneous atmosphere (worksheet #2, section 1.10)
in this case the dry rate is less