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Is the rate of heat transfer between two bodies brought into thermal contact dependent upon the different in temperature between them, assuming they each have the same mass? Imagine two bodies of water connected to each other by an impermeable glass barrier. If the difference between the temperatures is 10 degrees, will the system reach its equilibrium state faster per degree than if the difference is one degree?

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Yes, it is very much dependent on temperature gradient. The stefan-boltzman law is only for radiative dissipation of heat and not for general heat transfer by conduction(contact). The equation for steady state heat flow is $$\vec h=-k \vec{\nabla T}$$ where $\vec \nabla$ is the gradient operator which is based on the temperature difference between the bodies and $k$ is the thermal conductivity. $\vec h$ is the heat per unit area.

If the system is not in steady state, then the heat diffusion equation becomes relavent:- $$\frac{dT}{dt}=\frac{k}{c}\nabla^2 T$$ where $c$ is the specific heat of the body.
Lastly, a to a crude approximation, the rate of cooling in atmosphere at low difference of temperature of atmosphere and the body is given by Newton's law of cooling (partly by radiation):- $$\frac{dT}{dt}=-K(T_{body}-T_{surrounding})$$ where $K$ is a constant.

In all these equation, it is the temperature gradient between the bodies in contact, which drives the rate of heat transfer (or temperature change).

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Yes,

I believe according to Stefan-Boltzmann's law the hotter something is, then the more energy it will radiate per surface area, causing more energy to be lost faster at higher temperatures.

http://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law

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