Heat transfer vs calorimetry equation

Question

I'm trying to understand the difference between the equations $$Q=mc\Delta T$$ and, $$\frac{dQ}{dt}=\frac{-\kappa\alpha\Delta T}{l}$$

Suppose, we have two metal rods at $T_1$ and $T_2$ temperature, and we connect them at the end. I want to know, what do the two above equations tell us, regarding this system.

My understanding is that the first equation tells us, that the two systems will reach an equilibrium temperature, and the heat gained by one rod would be the heat lost by the other. This final temperature would depend upon the relative masses, and heat capacities. However, the first equation tells us, how much heat flows out of the first rod, into the second rod, and the final temperature.

Where does this second equation come from, and what does it tell us ? It seems to me, that the second equation would give us the heat transferred from one rod to the other, in some given time, unlike the first one that gives us the total heat transferred.

Combining the two above equations, we can derive the partial differential equation, called the heat equation. My question is, what does my heat equation tell me ? The first equation tells me, the final temperature of two bodies in contact, provided I know the initial temperature, mass and heal capacities.

What does the heat/diffusion equation tell me then ? Does it tell me, the distribution of heat at some given time ?

Does it mean, if I use the heat equation for our problem with the two rods at different temperatures, I'd get the same final temperature as the $Q=mc\Delta T$ equation, after some time, and this temperature would be constant ?

Answer 1

The first equation tells you, given an object with mass $m$ and specific heat $c$, how much heat you need to give to the system for it to change its temperature by $\Delta T = T_{final}-T_{initial}$.

The second equation instead describes the flow of heat over time in a system which is not at equilibrium and initially has a temperature distribution $T(x, y, z)$. Notice that now $\Delta$ is not a "difference" but a Laplace operator $\Delta = \partial_x^2+\partial_y^2+\partial_z^2 $.

Say you have a bar at temperature $T_{inital}$ and you start apllying heat to one side: over time the heat will flow through the bar. When you start apllying heat to it, it will then slowly equilibrate until it reaches $T_{final}$.

The first equation tells you the total heat necessary to heat the bar from the initial to the final temperature. The second one, will describe the process between the initial time and the final equilibration time.

In the case of the two bar, clearly the hot one will cool and the cold one will heat up until they both reach $T_{eq}$. In this case, the first equation tells you how much heat did the hot one give to the cold one. The second equation again would tell you how the heat has been flowing in the system over time until equilibrium.

If you do integrate the second equation over time it will in the end give you the total heat used, given by the first one.