One of the most important consequences of combining the First and Second Laws of Thermodynamics is that the resulting differential expressions reveal the variables that most naturally describe each thermodynamic function.
Recall that for a function of two variables,
\[ z=z(x,y) \]
the total differential is
\[ dz = \left( \frac{\partial z}{\partial x} \right)_y dx + \left( \frac{\partial z}{\partial y} \right)_x dy \]
The variables that appear in the differential are called the natural variables of the function.
For example, if
\[ dz = adx + bdy \]
then \(x\) and \(y\) are the natural variables of \(z\).
By comparing the differential forms of the thermodynamic functions to the general form of a total differential, the natural variables can be identified immediately.
Combining the First and Second Laws gives
\[ \boxed{ dU = TdS - pdV } \]
Comparing this expression with the general form of a total differential,
\[ dU = \left( \frac{\partial U}{\partial S} \right)_V dS + \left( \frac{\partial U}{\partial V} \right)_S dV \]
shows that the natural variables of \(U\) are
\[ \boxed{ U = U(S,V) } \]
Starting from
\[ H = U + pV \]
and differentiating,
\[ dH = dU + pdV + Vdp \]
substituting
\[ dU = TdS - pdV \]
gives
\[ \boxed{ dH = TdS + Vdp } \]
Therefore, the natural variables of enthalpy are
\[ \boxed{ H = H(S,p) } \]
Starting from
\[ A = U - TS \]
differentiation gives
\[ dA = dU - TdS - SdT \]
substituting
\[ dU = TdS - pdV \]
yields
\[ \boxed{ dA = -pdV - SdT } \]
Thus, the natural variables of the Helmholtz function are
\[ \boxed{ A = A(V,T) } \]
Starting from
\[ G = H - TS \]
differentiation gives
\[ dG = dH - TdS - SdT \]
substituting
\[ dH = TdS + Vdp \]
yields
\[ \boxed{ dG = Vdp - SdT } \]
Therefore, the natural variables of the Gibbs function are
\[ \boxed{ G = G(p,T) } \]
| Function | Differential | Natural Variables |
|---|---|---|
| \(U\) | \(dU = TdS - pdV\) | \(S,V\) |
| \(H\) | \(dH = TdS + Vdp\) | \(S,p\) |
| \(A\) | \(dA = -pdV - SdT\) | \(V,T\) |
| \(G\) | \(dG = Vdp - SdT\) | \(p,T\) |
Big picture: The natural variables of a thermodynamic function are the variables that appear in its total differential. Knowing the natural variables is important because they determine which partial derivatives arise naturally and ultimately lead to the Maxwell Relations.
Consider an ideal rubber band that can stretch but does not change volume. The work done on the rubber band is
\[ dw = -p\,dV + \tau\,dL \]
where \(\tau\) is the tension and \(L\) is the length of the rubber band.
Starting from the First Law,
\[ dU = dq + dw \]
and using \(dq_{\mathrm{rev}} = TdS\), we obtain
\[ dU = TdS - p\,dV + \tau\,dL \]
For an ideal rubber band, the volume does not change, so
\[ dV = 0 \]
Therefore,
\[ dU = TdS + \tau\,dL \]
Now define the Helmholtz function:
\[ A = U - TS \]
Differentiating gives
\[ dA = dU - T\,dS - S\,dT \]
Substituting the expression for \(dU\):
\[ dA = \left(TdS+\tau\,dL\right) - T\,dS - S\,dT \]
\[ dA = \tau\,dL - S\,dT \]
Since the differential contains \(dL\) and \(dT\), the natural variables of the Helmholtz function for this ideal rubber band are
\[ \boxed{ A = A(L,T) } \]
By inspection,
\[ \left(\frac{\partial A}{\partial L}\right)_T = \tau \]
\[ \left(\frac{\partial A}{\partial T}\right)_L = -S \]
Interpretation: For a simple compressible substance, the Helmholtz function is naturally written as \(A(V,T)\). For an ideal rubber band with no volume change, the relevant mechanical coordinate is not volume but length. The conjugate force to length is the tension \(\tau\), so the Helmholtz function is naturally written as \(A(L,T)\).
| Function | Differential | Natural Variables |
|---|---|---|
| \(U\) | \(dU = TdS - pdV\) | \(S,V\) |
| \(H\) | \(dH = TdS + Vdp\) | \(S,p\) |
| \(A\) | \(dA = -pdV - SdT\) | \(V,T\) |
| \(G\) | \(dG = Vdp - SdT\) | \(p,T\) |