What is the Error in My Derivation of the Dry Adiabatic Lapse Rate?

[jdlawlis]

I am aware of the derivation for the dry adiabatic lapse rate using the enthalpy approach: nc_pdT+VdP=0, but I can't seem to spot the error in my own derivation. If anyone sees it, I would be especially grateful.

dU=$$\delta$$Q-$$\delta$$W

nc_vdT=0 - PdV

$$\frac{dT}{dV}$$=-$$\frac{P}{nc_{v}}$$

$$\frac{dT}{dV}$$ $$\frac{dV}{dz}$$= -$$\frac{P}{nc_{v}}$$ $$\frac{dV}{dz}$$=-$$\frac{P}{nc_{v}}$$ $$\frac{dV}{dP}$$ $$\frac{dP}{dz}$$ by the chain rule

$$\frac{dT}{dz}$$ = $$\frac{-P}{nc_{v}}$$ $$\frac{-nRT}{P^{2}}$$ $$\frac{-MPg}{RT}$$ = $$\frac{-Mg}{c_{v}}$$

I obtained $$\frac{dV}{dP}$$ by differentiating PV = nRT and $$\frac{dP}{dz}$$ from the hydrostatic equation, substituting
density = mass/volume = MP/RT, where M is molar mass in g/mol.

For some reason, I obtain c_v in my derivation instead of the correct c_p.

Any thoughts?

 

[DEMcMillan]

You were kind to point out a defect in the Miskolczi model that I gather interferes with his virial theorem claim. I made another criticism of his and most other radiation balance models and hope that you will comment on it. https://www.physicsforums.com/showthread.php?t=261966
I noted your quandary and an absence of any reply. I offer this analysis to thank you for your Miskolczi comment that encouraged me to go further.

I spent some time on the web in order to identify past approaches to quantifying temperature change with altitude and noted more and more problems in approach. Your approach was to use a first law expression differentiated by dz as an altitude surrogate. You expected to need c_P but made a transcription error of your enthalpy statement into your first law conversion that left you with c_V . I want to identify a problem that is universal to all the websites that have attempted this quantitation. The c_P solution is cited from a 1980 textbook problem’s solution (reference 8, web citation 1). The ICAO (International Civil Aviation Organization) standard dry atmosphere table (Geigy Scientific Tables,1984, volume 3, pages 27-29) has been modestly revised over the years. Its development cites the perfect gas law, uses a plane parallel model, and gives a fall in temperature with altitude of 15-8.501 = 6.499 ^oC at 1,000 m, 8.501-2.004 = 6.497 ^oC at 2,000 m and 2.004- -4.491 = 6.495 ^oC at 3,000 m. The derivation of the temperature column is not made clear. The Wikipedia treatment gives us 9.8 ^oC/km, a rather higher number. http://en.wikipedia.org/wiki/Lapse_rate They use a poorly defined γ to get their rate.
http://pds-atmospheres.nmsu.edu/education_and_outreach/encyclopedia/adiabatic_lapse_rate.htm has posted a NASA approach that gives a value of 9.76 ^oC/km for Earth and shows that R = c_P – c_V. Underlying this approach and yours is the concept of specific heat capacity http://en.wikipedia.org/wiki/Specific_heat where c_P and c_V are in mass units and C_P (29.07) and C_V( 20.7643) are in molar units (dry air).

We should start with a reliable statement (. . . . used to maintain spacing, 2 attempted integral signs have a gap)
. . . . . _infinity
P_surface = ⌠-ρ g* dz with g* the altitude-affected gravitational constant less vector centrifugal force from body rotation
^. . . . . . . ⌡ . . . . . . . . . where P_infinity is zero and density area expands with body mass center radius squared
. . . . . ^surface
. . . . .. . . . . . . . . . . . . . . . ^infinity
which can be generalized to P = ⌠-ρ g* dz with no z directed mass transfer
^. . . . . . . . . . . . . . . . . . . . . . ⌡
. . . . . . . . . . . . . . . . . . . . ^altitude
whose derivative is dP/dz = -ρ g*
and from the perfect gas law, P = ρ RT M^-1, hence dP = ρ R M^-1 dT, then
dT/dz = -g* M R^-1 that becomes 34.163 ^oK/km, a value clearly too high. Using C_P instead of R we get 9.77 ^oK/km and C_V 13.68 ^oK/km. C_P is the most attractive but still 50% higher than the ICAO value of 6.50 ^oK/km. The C_P derived fall predicts an upper troposphere temperature well below that observed. We need to be concerned that the real needed thermal change estimate from C_P may be substantially higher than the currently accepted number because of the difference between 9.77 and 6.50 ^oK/km. Obviously 3/2 C_P would be a wonderful outcome. Is there a motion reason to make this claim for adiabatic expansion/contraction compared to heat capacity?

 

[D H]

I can't seem to spot the error in my own derivation. If anyone sees it, I would be especially grateful.

dU=$$\delta$$Q-$$\delta$$W

nc_vdT=0 - PdV

You are assuming a constant volume process on the left hand side (dU=nc_vdT) but a constant pressure process on the left hand side (dW=PdV). These two assumptions are at odds with one another.

 

[DEMcMillan]

jdlawlis wrote

For some reason, I obtain cv in my derivation instead of the correct cp.

I noted a second reply and was looking at the NASA dry adiabatic lapse rate citation again in developing a Venus atmosphere analysis. I noticed a feature that deals directly with your problem. The NASA solution started as you did the second time with c_V in the first law expression and then substituted c_P – c_V for R in the gas law equation to allow n c_V dT – n c_V dT to disappear, leaving only c_P in the adiabatic expression. You still have the problem of a 3/2 too large result, but it was not a simple transcription error as I first thought.

 

[DEMcMillan]

I have spent some time looking into the origin of the ICAO table and sense that the lapse rate there is compromised by specific aviation needs http://www.netweather.tv/forum/index.php?showtopic=49857&pid=1341485&st=0&#entry1341485 . c_P is appropriate, not 3/2 c_P. The NASA approach takes the derivative directly, while you omitted the T component of the derivative (dT/dP) part in your dV/dP ( n R/P ^. dT/dP – n R T/ P²). Recovery by substitution is possible, but I found no easy path.
The most direct approach is
dQ = dU + δW = n^.c_V dT+ P dV = 0 , first law
V dP + PdV = n R dT
dQ = n c_V dT - V dP + n (c_P – c_V) dT
dQ = n c_P dT - V dP = 0 for adiabatic, V = n <mw> / ρ
dT/dz = <mw> dP/dz / ρ c_P, where dP/dz = - g^* ρ
dT/dz = - g^* <mw> /c_P = - g^* /C_P

I suspect we don’t usually see the predicted low upper tropospheric temperatures because strong clear air downdrafts generate considerable g^*-lowering mass transfer associated with high local barometric pressure and horizontal air movement away from the surface site.