The rate of energy transfer between parallel flat plates is evaluated when the (stagnant) gas between them is polyatomic with one inert internal mode. Deviations of the thermal conductivity from the complete equilibrium (Eucken) value are expressed in terms of the inert mode relaxation time and the effectiveness of the walls in exciting or de-exciting this mode. The results are obtained via a linear theory consistent with small temperature differences between the plates. It is found that the Eucken-value of conductivity could be exceeded if the relaxation times are non-zero and the plates very effective in exciting the inert mode. 1Nhen relaxation times are very short the effect of the walls on the energy transfer rate is small, but the malls make their presence felt by distorting the temperature profiles in "boundary layers" adjacent to the walls which are of order VDT in thickness (D = diffusion coefficient, r = relaxation time). This result is analogous to Hirschfelderts (1956) for the case of chemical reactions. For experimental measurement of conductivity in a hot wire cell type of apparatus it is shown that extrapolation of measured reciprocal conductivities to zero reciprocal pressure should lead to the full Eucken value. It is also shown that the slope of reciprocal apparent (measured) conductivity versus reciprocal pressure curves is a function of relaxation time as well as of the accommodation coefficients. It is quite possible that the relaxation effect here is comparable with the temperature jump effects, even for rotation in diatomic molecules.