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Chemical Thermodynamics 2013/2014. 4 th Lecture: Manipulations of the 1 st Law and Adiabatic Changes Valentim M B Nunes, UD de Engenharia. Relations between partial derivatives.

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Chemical thermodynamics 2013 2014


4th Lecture: Manipulations of the 1st Law and Adiabatic Changes

Valentim M B Nunes, UD de Engenharia

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

Relations between partial derivatives

The Inverter:

The Permuter:

Euler’s chain relation:

Finally, the differential df = g dx + h dy is exact, if:

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:

Change of U with T at constant pressure

What does this mean?

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!

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?

Change of H with T at constant volume

What does this mean?

Using the Euler’s relation


We define now the isothermal compressibility coefficient

To assure that kT is positive!

So, we find that

Change of H with T at constant volume

Using again the Euler’s relation and rearranging


What is this? See next slide! For now we will call it µJT

We finally obtain

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!

The Joule-Thomson effect

For an ideal gas, µJT = 0. For most real gases Tinv >> 300 K. If µJT >0 the gas cools upon expansion (refrigerators). If µJT <0 then the gas heats up upon expansion.

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!

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!

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,

Adiabatic vs isothermal expansion