Understanding Specific Heat: A Derivation of the Correct Equation

[FAS1998]

Homework Statement

1.2.6. Supose that the specific heat is a function of position and temperature, c(x,u).

(a)Show that the heat energy per unit mass necesary to raise the temperature of a thin slice of thickness deltax from 0°to u(x,t) is not c(x)u(x,t), but instead int((0->u)c(x,u’))du’.

Homework Equations

The Attempt at a Solution

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I believe c(x,u) gives the energy per unit mass required to raise the temperature by 1 K as a function in of position and temperature.

And u(x,t) is the temperature as a function of position and time.

So I would’ve thought that the energy per unit mass required to raise the temperature to u(x,t) would be the desired temperature (u(x,t)) multiplied by the specific heat (c(x,u(x,t))).

This would give me u(x,t)c(x,u(x,t))), which is not what they’re looking for, and is very close to what they explicitly say is an incorrect answer.

 

[Orodruin]

The heat capacity is not the energy required to raise the temperature by 1 K. It is the energy required to raise the temperature a small amount divided by that small change in temperature. In other words, it is a derivative of internal energy wrt temperature. The multiplication will only hold if the heat capacity is constant with temperature.

Also, pet peeve, there is nothing called ”degrees Kelvin”. The unit of temperature is just ”Kelvin”, nothing else.

 

[mjc123]

I believe c(x,u) gives the energy per unit mass required to raise the temperature by 1 degree Kelvin as a function in of position and temperature.

When c varies with temperature, it is better to define it by saying that c(x,u)du is the energy required per unit mass to raise the temperature from u to u+du. Now the value of c is different, because the temperature is different. To increase temperature by another du requires heat c(x, u+du)du. Adding these steps together, from the initial to the final temperature, gives the integral you require. (I think you meant to say int(0→u), not int(u→u))
You cannot simply multiply c by the temperature interval u when c varies with temperature.