= Solution
For the <thermoelectric couple figure of merit>, the coefficient $\alpha_{np}$ denotes a \b[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. Then
$$
R=2\rho L/A,\qquad K=2\kappa A/L,\qquad
\boxed{Z_{\rm pair}=\frac{\alpha_{np}^2}{RK}=\frac{\alpha_{np}^2}{4\rho\kappa}=3.83\times10^{-4}\,\mathrm{K^{-1}}.}
$$
This 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 $Z_{\rm pair}\overline T=0.162$.
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 \b[$Z=1.53\times10^{-3}\,\mathrm{K^{-1}}$], 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.
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