[G O D I V A]

Here's the question:

Calculate the change in (a) the molar internal energy and (b) the molar enthalpy of liquid water when its temperature varies by 10 K.  Account the difference between the two quantities.

C_{p, m} (298K) = 75.3 JKmol
C_{v, m} (298K) = 74.8 JKmol

It shows me what to do but I dont understand why its doing it.  For example, for (a) it says

dU_m = C_{v, m}dT .(then integrates it, subs its values). why is this?

I thought dU = (dU/dV)_TdV + (dU/dT)_VdT.

So what happened to the (dU/dV)_TdV?  Am i supposed to assume constant pressure?

[renge ishyo]

The equation dU_m = C_v,mdT holds for constant volume and constant moles. The subscripts V and m indicate for which terms the amounts are held constant for.

From the full formal equation the (dU/dV)_TdV term cancels out to zero when the volume is constant (dV= 0), so that the equation simplifies to:

dU = (dU/dT)_VdT

From this you can see that:

C_v = (dU/dT)_V

Or in other words, the heat capacity "C" is just a slope, a rate of change of energy with respect to temperature. The higher the heat capacity C, the more energy that is absorbed for a given increase in temperature.

The reason why you have to use separate heat capacities with constant V (for internal energy) or constant P (for enthalpy) is to cancel out one of the two terms in the formal equation (as done above for the full equation where dV=0 allowed a simplification to just one term). This allows you to solve the problem. You can use C_v to get the change in internal energy (so use C_v for part A) at constant volume, and you can use C_p to get the change in enthalpy (so use C_p for part B) at constant pressure.

[leela sriram]

the value of (dU/dV)_{T = 0 for ideal gases
and for non ideal gases it is not equal to zero
this is called JOULES coeffiecient of free expansion of gas}