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Department of Atmospheric SciencesATM 404/504Thermodynamics

Ch. 41st Law of Thermodynamics

Heat Capacities

o Consider a homogeneous system of constant compositiono Write dU(dH) as a total differential of the independent

variablesXand Y(for our purposes,Xand Ycould be any

pair ofp, V, and T). Then

o From the 1st Law, Q = dU+pdV, we can substitute to get

whereXand Yare Tand V, respectively.

( ) ( )Y X

dU U X dX U Y dY = +

( ) ( )V T

Q U T dT p U V dV d = + +


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Heat Capacities

o From Q = dHVdp we can substitute fordHto get

whereXand Yare Tandp, respectively.

o For a process of heating at V= const we get

or per unit mass we can write

( ) ( )p T

Q H T dT H p V dpd = + -

( )v v Vc m C Q dT U T d= = =

( )vv

c q dT u T d= =


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Heat Capacities

o For a process of heating atp = const we get

or per unit mass we can write

o Can also calculate heat of change ofVandp at T= const

o Not very useful, but given for completeness

( )p p pc m C Q dT H T d= = =

( )p pc q dT h T d= =

( ) ( ), andT T

Q dV U V p Q dp H p V d d= + = -


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Calculation of Internal Energy and Enthalpy

o Our equations forCVand Cp can be integrated directly forprocesses with V= const andp = const, respectively to

find UandH, ifCVand Cp are known as functions ofT

oCVand Cp, from experiment, are usually polynomials in Tandv pU C dT const H C dT const

= + = +

2C T Ta b g= + + + K


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Calculation of Internal Energy and Enthalpy

o Consider 2 rigid vessels linked by a connection with astopcock (pg. 34). One contains gas, the other evacuated.

o Stopcock opened, gas in 1 expands to occupy both vessels.

o Temperature measurements show that system exchanges

no heat with environment no work done, Q = 0, W= 0,

and U= 0.

o Sincep changed during process, we have U= U(T) only

and partial derivative used above are total derivatives.


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Ch. 41st Law of ThermodynamicsCalculation of Internal Energy and Enthalpy

o Notes on this experiment

When experiment done carefully, small heat exchange was found(Joule-Thomson effect), which vanishes for ideal gas behavior

As gas confined in vessel 1 expands into 2, work is done by someportions of gas against others while volumes change (as molecules enter2, they are effected by molecules following)

These are internal transfers that are not included in W

This shows the importance of defining system carefully and clearlywhen considering a thermodynamic process

In this case, the system is all the gas contained in both vessels (initially

one is empty), whose total volume (V1 + V2) does not change


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More on Heat Capacities

o As noted above, since U= U(T) we have CV= dU/dTandcv = du/dT

o We can writeH= U+pV= U+ nR*T=H(T) leading to

Cp = dH/dTand cp = dh/dT

oSince we are only interested in differences in internal

energy and enthalpy, we can set the integration

constant to 0 giving: UCVT,HCpT, ucvT, and

hcpT


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o Since we have CV= dU/dT, Cp = dH/dTandH= U+pV=U+ nR*T, we have

leading to Cp

CV

= nR*cp

cv

=R recalling that

n = m/M.

oAs mentioned earlier, heat capacities for all gases can

be measured and the coefficients for the polynomial

expansion can be determined (C= + T+ T2+ )

* *

p VdH dT dU dT nR C C nR= + = +


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Ch. 41st Law of ThermodynamicsMore on Heat Capacities

o For simple gases like N2, O2, and Ar, the experimental data are

nearly constant for all temperatures and pressures of interest, so the

temperature variation is not considered.

o From earlier we have, for monatomic gases, the total internal energy

is U= (3/2)NkT, which leads to CV= (3/2)nR* and cv = (3/2)R.

Similarly, Cp = (5/2)nR* and cp = (5/2)R.

o For diatomic gases, where there are more degrees of freedom, so we

get CV= (5/2)nR* and cv = (5/2)R. Similarly, Cp = (7/2)nR

* and cp =

(7/2)R. The ratios Cp/CV= cp/cv = .


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o Dry air is considered to be a diatomic gas, so the secondform applies.

o The ratio, cp/cv = = 1.4.

o We then attach the subscript, d, to the specific heats to

indicate dry air.

o This leads to cvd= 718 J kg-1 K-1 and cpd= 1005 J kg

-1 K-1

andRd= cpdcvd= 287(.05) J kg-1 K-1.


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Ch. 4

1

st

Law ofThermodynamics


o The table below shows the values ofcpdfor varioustemperatures and pressures. Note the slight variation.


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More Forms of the 1st Law

o Using the above expressions for (specific) heat capacities,we get more useful forms of the 1st Law, two of which are

particularly useful

and

, andv vQ C dT pdV q c dT pd d d a= + = +

, andp pQ C dT Vdp q c dT dpd d a= - = -


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Special Cases

o For an isothermal transformation

o For an isochoric (constant volume) transformation

o For an isobaric transformation

*ln f iQ nR T V V =

(V f iU Q C T T D = = -

( ) (

( ) ( )

andp f i f i

p f i f i

Q C T T W p V V

U C T T p V V

= - = -

D = - - -

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