Hello sir, sorry to bother you again, but I had another question in case you're not busy

What's the difference between the calorimetry equation Q=mc dT, and the heat equation T_{t}=kT_{xx} ? If we know, the final temperature using the calorimetry equation, what exactly does the heat differential equation tell us ?

As in, if I have two rods of the same type at two different temperatures, but same mass and heat capacity , and I bring them in contact at the two ends, using the calorimetry equation, I can figure out the final temperature, which would just be the average of the initial temperatures in this case.

Hi Ray :-)

Using the heat equation, I guess we are finding the temperature distribution at each point in time, because, this final temperature is not reached instantaneously. However, using the heat diffusion equation, at time tending to infinity, will I obtain the exact same final temperature as I did, using the calorimetry equation ?

So we'd use the calorimetry equation when we only care about the final result and not about how the temperature evolved with time.

okay, and in case of suppose a single rod, heated at one end for a while, such that the two ends are at different initial temperatures, I think I can use the heat equation to get the temperature profile at any instant, and also the final temperature.

But is the calorimetry equation useful in this case, when we have a single body ?

If you have a single isolated body then no heat flows in or out of it, so the average temperature is constant.

The calorimetry equation would just tell you the change in the average temperature is zero.

Which probably isn't that useful :-)

Ah so, the calorimetry equation just tells me the final average temperature between two or more bodies with different average temperatures

Yes

the heat equation tells the temperature of any body at any point on the body, at any time

as in, if I connect the bodies, and draw a graph of temperatures, it will give me how this graph changes , and what it looks like at a certain time, provided we know how it looked like in the beginning

Thank you so much, again sir, you have been a life saver :D

You're welcome :-)