05.04.2016 Views

Modern Engineering Thermodynamics

You also want an ePaper? Increase the reach of your titles

YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.

11.4 Maxwell Equations 367<br />

11.4 MAXWELL EQUATIONS<br />

Two sets of equations are named after the Scottish physicist James Clerk Maxwell (1831−1879): the electromagnetic<br />

field equations and the thermodynamic property equations. The thermodynamic Maxwell equations allow<br />

additional numerical information to be obtained about the nonmeasurable properties u, h, and s from accurately<br />

measured p, v, and T data.<br />

Consider an arbitrarily continuous function of the form<br />

Then, we can write its total differential as<br />

where we set<br />

dz =<br />

<br />

∂z <br />

∂x y<br />

dx +<br />

z = zx, ð yÞ<br />

<br />

∂z<br />

<br />

dy = Mdx + Ndy (11.11)<br />

∂y<br />

x<br />

<br />

M =<br />

∂z <br />

∂x y<br />

and<br />

<br />

N =<br />

∂z<br />

<br />

∂y<br />

x<br />

If we now differentiate M with respect to y while holding x constant and differentiate N with respect to x while<br />

holding y constant, we get<br />

<br />

∂M<br />

= ∂2 z<br />

∂y<br />

x<br />

∂y ∂x<br />

and<br />

<br />

∂N<br />

= ∂2 z<br />

∂x y ∂x ∂y<br />

Since we require z(x, y) to be a continuous function, it follows that<br />

or that<br />

∂ 2 z<br />

∂y ∂x =<br />

<br />

∂M<br />

∂y<br />

x<br />

∂2 z<br />

∂x ∂y<br />

<br />

≡ ∂N <br />

∂x y<br />

(11.12)<br />

Recall that the thermodynamic state of any pure substance is fixed by any pair of independent intensive thermodynamic<br />

properties of that substance. That is, any property of a pure substance can be written as a function of<br />

any other two independent properties of that substance. Consequently, if x and y are such independent properties,<br />

then z also is a property, provided that Eq. (11.12) is satisfied.<br />

EXAMPLE 11.4<br />

Suppose we make a series of measurements in the laboratory and think we discovered a new thermodynamic property, call<br />

it z. Our experimental data provide an empirical equation of the form: dz = pdv + v 2 dp. Isz a new property?<br />

Solution<br />

The unknown is whether or not z is a new thermodynamic property. Equation (11.11) here has the form<br />

dz = Mdx + Ndy = pdv+ v 2 dp<br />

so M = p, N = v 2 , x = v, and y = p. The cross differentials in Eq. (11.12) are<br />

<br />

∂M<br />

= ∂p <br />

= 1<br />

∂y<br />

x<br />

∂p<br />

v<br />

(Continued )

Hooray! Your file is uploaded and ready to be published.

Saved successfully!

Ooh no, something went wrong!