Chemical Thermodynamics 2018/2019

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Chemical Thermodynamics2018/2019
4th Lecture Manipulations of the 1st Law and
Adiabatic Changes Valentim M B Nunes, UD de
Engenharia
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Relations between partial derivatives
Partial derivatives have many useful properties,
and we can use it to manipulate the functions
related with the first Law to obtain very useful
thermodynamic relations. Let us recall some of
those properties.
If f is a function of x and y, f f(x,y), then
If z is a variable on which x and y depend, then
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Relations between partial derivatives
The Inverter
The Permuter
Eulers chain relation
Finally, the differential df g dx h dy is
exact, if
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Changes in Internal Energy
Recall that U U(T,V). So when T and V change
infinitesimally
The partial derivatives have already a physical
meaning (remember last lecture), so
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Change of U with T at constant pressure
Using the relation of slide 2 we can writhe
We define the isobaric thermal expansion
coefficient as
Finally we obtain
Closed system at constant pressure and fixed
composition!
0 for an ideal gas
Proofs relation between Cp and Cv for an ideal
gas!
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Change of H with T at constant volume
Let us choose H H(T,p). This implies that
Now we will divide everything by dT, and impose
constant volume
What is the meaning of this two partial
derivatives?
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Change of H with T at constant volume
Using the Eulers relation
Rearranging
We define now the isothermal compressibility
coefficient
To assure that kT is positive!
So, we find that
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Change of H with T at constant volume
Using again the Eulers relation and rearranging
or
What is this? See next slide! For now we will
call it µJT
We finally obtain
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The Joule-Thomson Expansion
Consider the fast expansion of a gas trough a
throttle
If Q 0 (adiabatic) then
So, by the definition of enthalpy
Isenthalpic process!
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The Joule-Thomson effect
For an ideal gas, µJT 0. For most real gases
Tinv gtgt 300 K. If µJT gt0 the gas cools upon
expansion (refrigerators). If µJT lt0 then the gas
heats up upon expansion.
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Adiabatic expansion of a perfect gas
From the 1st Law, dU dq dw. For an adiabatic
process dU dw and dU CvdT, so for any
expansion (or compression)
For an irreversible process, against constant
pressure
The gas cools!
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Adiabatic expansion of a perfect gas
For a reversible process, CVdT -pdV along the
path. Now, per mole, for an ideal gas, PV RT, so
For an ideal gas, Cp-Cv R, and introducing
then
The gas cools!
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Adiabatic expansion of a perfect gas
We can now obtain an equivalent equation in terms
of the pressure
As a conclusion, is constant
along a reversible adiabatic.
For instance, for a monoatomic ideal gas,
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Adiabatic vs isothermal expansion