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Integrating Factors, Exact Forms

Exact Differential Forms

In classical thermodynamics for simple fluids, expressions such as

$$\begin{split}dU = & T dS - P dV\ = & \ensuremath{ \left( \fr... ...tial{V}} \right)_{S} } dV\ = & \delta q + \delta w \end{split}$$ (20-8)

represent the differential form of the combined first and second laws of thermodynamics. If $dU=0$ , meaning that the differential Eq. 20-8 is evaluated on a surface for which internal energy is constant, $U(S,V) =$   const , then the above equation becomes a differential form

$$0 = \ensuremath{ \left( \frac{\partial{U}}{\partial{S}} \right)_{V} } dS + \ensuremath{ \left( \frac{\partial{U}}{\partial{V}} \right)_{S} } dV$$ (20-9)

This equation expresses a relation between changes in $S$ and changes in $V$ that are necessary to remain on the surface $U(S,V) =$   const .

Suppose the situation is turned around and you are given the first-order ODE

$$\ensuremath{\frac{d{y}}{d{x}}} = -\frac{M(x,y)}{N(x,y)}$$ (20-10)

which can be written as the differential form

$$0 = M(x,y) dx + N(x,y) dy$$ (20-11)

Is there a function $U(x,y) =$   const or, equivalently, is it possible to find a curve represented by $U(x,y) =$   const ?

If such a curve exists then it depends only on one parameter, such as arc-length, and on that curve $dU(x,y)=0$ .

The answer is, ``Yes, such a function $U(x,y) =$   const exists if an only if $M(x,y)$ and $N(x,y)$ satisfy the Maxwell relations''

$$\ensuremath{\frac{\partial{M(x,y)}}{\partial{y}}} = \ensuremath{\frac{\partial{N(x,y)}}{\partial{x}}}$$ (20-12)

Then if Eq. 20-12 holds, the differential form Eq. 20-11 is called an exact differential and a $U$ exists such that $dU = 0 = M(x,y) dx + N(x,y) dy$ .

Integrating Factors and Thermodynamics

For fixed number of moles of ideal gas, the internal energy is a function of the temperature only, $U(T) - U(T_o) = C_V (T - T_o)$ . Consider the heat that is transfered to a gas that changes it temperature and volume a very small amount:

$$\begin{split}dU = & C_V dT = \delta q + \delta w = \delta q - P dV\ \delta q & = C_V dT + P dV \end{split}$$ (20-13)

Can a Heat Function $q(T,V) =$   constant be found?

To answer this, apply Maxwell's relations.