# 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? Mar 1 '19 at 11:57
• that's my question. Mar 1 '19 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. Mar 1 '19 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 Mar 1 '19 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. Mar 1 '19 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 Mar 1 '19 at 11:23
• It's my downvote. Down votes are for answers that are "not useful" to answering the question. Mar 1 '19 at 12:21