Specific Heat Capacity Derivation

[cwill53]

Homework Statement

Problem with Specific Heat Capacity Derivation

Relevant Equations

$c_v=\left ( \frac{\partial u}{\partial T} \right )_v$
$c_p=\left ( \frac{\partial h}{\partial T} \right )_p$

The specific heat capacity at constant volume and the specific heat capacity at constant pressure are intensive properties defined for pure, simple compressible substances as partial derivatives of the functions u(T, v) and h(T, p), respectively,
$$c_v=\left ( \frac{\partial u}{\partial T} \right )_v$$
$$c_p=\left ( \frac{\partial h}{\partial T} \right )_p$$
Can someone explain why this is? The book from which I got this from doesn't derive these expressions. I want to know what expressions for u(T,v) and h(T,p) are differentiated.

 

[Chestermiller]

Homework Statement:: Problem with Specific Heat Capacity Derivation
Relevant Equations:: $c_v=\left ( \frac{\partial u}{\partial T} \right )_v$
$c_p=\left ( \frac{\partial h}{\partial T} \right )_p$

The specific heat capacity at constant volume and the specific heat capacity at constant pressure are intensive properties defined for pure, simple compressible substances as partial derivatives of the functions u(T, v) and h(T, p), respectively,
$$c_v=\left ( \frac{\partial u}{\partial T} \right )_v$$
$$c_p=\left ( \frac{\partial h}{\partial T} \right )_p$$
Can someone explain why this is? The book from which I got this from doesn't derive these expressions. I want to know what expressions for u(T,v) and h(T,p) are differentiated.

They are impossible to derive...because they are definitions. They will match the old definitions you are used to, in terms of heat, in the specific cases of constant volume and constant pressure, respectively. But defining them in terms of heat is not really valid because they are physical properties of the material, and not related to heat, which is process path dependent (and thus can't be a physical property).

 

[cwill53]

They are impossible to derive...because they are definitions. They will match the old definitions you are used to, in terms of heat, in the specific cases of constant volume and constant pressure, respectively. But defining them in terms of heat is not really valid because they are physical properties of the material, and not related to heat, which is process path dependent (and thus can't be a physical property).

I see. Can heat capacity in general, not specific heat, be defined in terms of physical properties? Or are those also definitions.

 

[Chestermiller]

I see. Can heat capacity in general, not specific heat, be defined in terms of physical properties? Or are those also definitions.

My experience is that most of the time, people use the terms specific heat and heat capacity synonymously. However, in some cases, heat capacity is used to mean specific heat times the number of moles or mass. So, either way, they are defined the way I have said. And in the latter case, heat capacity is the extensive property equivalent of specific heat, just as U is the extensive property equivalent of u.

 

[cwill53]

My experience is that most of the time, people use the terms specific heat and heat capacity synonymously. However, in some cases, heat capacity is used to mean specific heat times the number of moles or mass. So, either way, they are defined the way I have said. And in the latter case, heat capacity is the extensive property equivalent of specific heat, just as U is the extensive property equivalent of u.

Thanks for the informative reply!