= Kelvin thermoelectric relation
{c}
{title2=$\Pi=TS$}
With the <Seebeck coefficient> convention $dV=-S\,dT$ at zero <electric current>, and the transported-<heat> convention of <Peltier coefficient>, the <Kelvin thermoelectric relation> is
$$
\Pi(T)=T S(T),\qquad
\Pi_{A\to B}^{\rm abs}(T)=T\bigl[S_B(T)-S_A(T)\bigr].
$$
$T$ is absolute <temperature> in <kelvins>; use consistent material references. This scalar <linear response> relation assumes local <thermal equilibrium> and reciprocal transport without time-reversal-breaking fields or order. <Onsager reciprocal relations> supply the symmetry; <conservation of energy> alone is insufficient. A <magnetic field> requires a reversed-field reciprocity comparison.<Thermoelectric effects by Leon van Dommelen>[A.11.6: reciprocity and the Peltier–Seebeck relation]{ref}
In particular, the A-minus-B <thermocouple> coefficient in <Seebeck effect> is $S_A-S_B$, whereas absorbed junction <heat> for <conventional current> A to B uses $S_B-S_A$. There is no contradiction: terminal voltage order and crossing direction are separate choices. Multiplying a measured voltage by an absolute <temperature> without specifying these choices does not determine junction heating.
A short reciprocity calculation makes the factor $T$ visible. At a common reference <temperature>, write the local <linear response> with electrical driving field $\mathcal E=-\nabla V$ and <heat flux> $q$ as
$$
j=L_{11}\frac{\mathcal E}{T}+L_{12}\frac{-\nabla T}{T^2},\qquad
q=L_{21}\frac{\mathcal E}{T}+L_{22}\frac{-\nabla T}{T^2}.
$$
At $j=0$, the <Seebeck coefficient> convention gives $S=L_{12}/(T L_{11})$. At $\nabla T=0$, the <Peltier coefficient> is $q/j=L_{21}/L_{11}$. Thus $L_{21}=L_{12}$ gives $\Pi=TS$. The chosen forces are conjugate to the fluxes in the local <entropy production>, $j\mathcal E/T+q(-\nabla T)/T^2$; arbitrary rescaling of just one force would spoil this argument.
The relation is local. Varying $S(T)$ permits distributed <Thomson effect> heating, with $\tau=T\,dS/dT$; the constant-coefficient model below omits it. Endpoint substitutions into a constant-property formula can therefore break its <energy> accounting.
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