What is the Relationship Between Heat Energy and Temperature?

[tavrion]

Homework Statement

Show that the heat energy per unit mass necessary to raise the temperature of a thin slice of thickness \Deltax from 0^o{} to u(x,t) is not c(x)u(x,t). but instead \int_0^uc(x,\overline{u})d\overline{u}.

Homework Equations

According to the text, the relationship between thermal energy and temperature is given by

e(x,t) = c(x)p(x)u(x,t),

which states that the thermal energy per unit volume equals the thermal energy per unit mass per unit degree times the temperature time the mass density.

When the specific heat c(x) is independent of temperature, the heat energy per unit mass is just c(x)u(x,t).

The Attempt at a Solution

The only hint really is that this is related to the area, from the solution. How can I go about this geometrically and/or algebraically?

Any help/pointers will be much appreciated. Thank you!

 

[lanedance]

that integral doesn't make sense based on what you've posted, can you check it... c has changed to a function of x only or x and u (position and temperature)...?

 

[lanedance]

so i think you do need to assume c = c(x,u), then try and find the change in energy de, for a small change in temp du and integrate.

 

[tavrion]

Argh of course! Thank you very much!

 

[tavrion]

I have used your advice and went about it in the following way:

For a small slice of thickness \Delta{x} a small change in energy will be given by

de = c(x,u)du

Dividing by du I obtain e_{u} = c(x,u).

From the Fundamental Theorem of Calculus, this really says that

e(x,t) = \int_0^uc(x,t)dt.

Am I correct in my thinking? It feels a bit messy somehow...

 

[lanedance]

you can just start from differentials
de = c(x,u)du
\delta e = \int_{e_0}^{e_f}d\bar{e} = \int_{0}^{u} c(x,\bar{u})d\bar{u}

 

[tavrion]

Your equation did not even occur to me but simplifies it a lot. Thank you very much for your help. I can now get some sleep again :-)