# Change of specific internal energy due to temperature and volume

The change of internal energy due to temperature and volume may be written as $$dU=C_V dT + \left( T\left(\frac{\partial S}{\partial V}\right)_T-P\right) dV$$ where $$\left(\frac{\partial S}{\partial V}\right)_T=\left(\frac{\partial P}{\partial T}\right)_V.$$

In hydrodynamics you usually have to deal with thermodynamic quantities which are formulated per unit mass like the specific enthalpy or the specific internal energy, the specific volume and so forth.

I was wondering if the above equations are still valid if I naively replace the corresponding quantities with their "specific equivalents or does it give rise to additional terms?

• You can't just divide both sides of the equation by the mass? Commented Mar 1, 2019 at 11:57
• that's my question. Commented Mar 1, 2019 at 12:00
• I mean if that puts the equation in terms of values you need, then go for it. Unless your mass isn't constant, then the step probably wouldn't be valid anymore. Commented Mar 1, 2019 at 12:02
• Well that's the problem. The volume changes and so does the mass (specific volume= 1/density). But the above equation takes care of volume-change by means of the second term. Without change in volume I'd just need c_V*dT Commented Mar 1, 2019 at 12:18
• If you're asking whether U, Cv, and V can be either per mole of per unit mass, the answer is "yes" provided all three quantities have consistent units. Commented Mar 1, 2019 at 12:31

In the right hand side of the equation fo $$dU$$ you have an extensive quantity $$dV$$ which can be divided by either $$N$$ or $$V$$ without problems (the same for $$dU$$), and a couple af derivatives, $$C_V$$ and $$\left(\frac{\partial{S}}{\partial{V}}\right)_T$$. Derivatives are potentially a problem because, in order to get specific quantities one has to interchange derivative and division by the relevant extensive quantity.

In this case there is no problem, because $$C_V$$ is $$C_V = T\left(\frac{\partial{S}}{\partial{T}}\right)_{V,N}$$ and $$\left(\frac{\partial{S}}{\partial{V}}\right)_{T}= \left(\frac{\partial{S}}{\partial{V}}\right)_{T,N}$$ i.e. both partial derivatives have to be done at fixed $$N$$. Therefore, division by the total mass would imply division by $$M=Nm$$, where m is a constant and exchange of partial derivatives and division by $$M$$ is allowed without extra-terms.

The equations are independent of the units - no extra terms needed.

The word "specific" means by "per unit mass" - typically measured in $$J/kg\cdot K$$

The word "mole" means a "unit of substance" - each substance has a different mass - typically measured in $$J/mole\cdot K$$

If you use specific heat capacities, then different substances have no interesting regularities.

If you use molar heat capacities, then different substances have very interesting regularities.

• I know what it means but its not answering the question Commented Mar 1, 2019 at 11:23
• It's my downvote. Down votes are for answers that are "not useful" to answering the question. Commented Mar 1, 2019 at 12:21