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 $\Delta$x 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 :-)