Consider two slabs - Slab A and Slab B, insulated on "latereal" faces as shown, initially at the same temperature, and having identical dimensions. The slabs at t= 0 are brought in contact with two heat reservoirs (on left and right) at temperatures $T_1$ and $T_2$. Slabs have the same thermal conductivity but different specific heats, with $c_A > c_B$
Since specific heat of A > that of B I argue that the temperature profiles at any instant of time t, would be as follows:
i.e. since $c_A > c_B$ A will have a hard time raising it's temperature than B. As a result the temperature gradients in A will be smaller (in magnitude) than in the case of B. This would mean that the heat transferring to A from the left reservoir in any time dt is smaller in A than in B. Furthermore, the rate of heat transfer in intermediate layers will also be lower in A than B. I've often read that a higher specific heat restricts thermal diffusion, could this be one way of explaining it why?