1 / 14

# Chemical Thermodynamics 2013/2014 - PowerPoint PPT Presentation

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.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about ' Chemical Thermodynamics 2013/2014' - orien

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### ChemicalThermodynamics2013/2014

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

Valentim M B Nunes, UD de Engenharia

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

The Inverter:

The Permuter:

Euler’s chain relation:

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

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:

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!

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?

What does this mean?

Using the Euler’s relation

Rearranging

We define now the isothermal compressibility coefficient

To assure that kT is positive!

So, we find that

Using again the Euler’s relation and rearranging

or

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

We finally obtain

Consider the fast expansion of a gas trough a throttle:

If Q = 0 (adiabatic) then

So, by the definition of enthalpy

Isenthalpic process!

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.

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!

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!

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,