Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2022 / Paper 1 / Section B / 12 / iii / a

For the thermoelectric couple figure of merit, the coefficient ${\alpha }_{np}$ denotes a pair difference, whereas the table gives one value each for resistivity and thermal conductivity without saying whether they are per-leg or effective pair values. To keep the factors consistent, first take both legs to have the listed $\rho$ and $\kappa$, and equal $A,L$ as stated. ThenThis is identical to treating each symmetric leg as having Seebeck-coefficient magnitude ${\alpha }_{np}/2$. The mean temperature is $\overline{T}=(523.15+323.15)/2=423.15\mathrm{K}$, giving $Z_{\mathrm{pair}}\overline{T}=0.162$.

If the examination instead intends its $\rho$ and $\kappa$ as already combined effective pair coefficients in $R=\rho L/A$ and $K=\kappa A/L$, the alternative convention gives $Z=1.53\times 10^{-3}{\mathrm{K}}^{-1}$, with $Z\overline{T}=0.649$. That convention must be declared; using the full pair Seebeck difference with individual-leg $\rho ,\kappa$ while omitting both factors of two would overestimate the device figure of merit by four.