# Solids and Liquids: Internal Energy and Enthalpy

Both liquids and solids can be considered incompressible substances.  As a result, their specific volume will remain constant during a process.  This means that any energy associated with a volume change is  considered negligible in comparison to other forms of energy.  Due to this fact, the specific heat $c_p$ and $c_v$ will be equal to each other for a liquid and a solid.  Hence, $c_p=c_v=c$.  In turn, this will effect how the internal energy and enthalpy are calculated.

### Internal Energy: Solids and Liquids

Similar to an ideal gas, the internal energy of a liquid and solid is dependent on temperature only.  As a result, the partial differential equation in relation to $c_v$ will be as follows.

(Eq 1) $du=c_vdT=c(T)dt$

For a process, the change in internal energy from state 1 to state 2 is obtained using the following integration.

(Eq 2) $Δu=u_2-u_1=\int{}^2_1~c(T)dT$  (kJ/kg)

Finally, for small temperature differences, the average specific heat $c$ can be used to determine the change in internal energy.

(Eq 3) $Δu≅c_{avg}(T_2-T_1)$  (kj/kg)

### Enthalpy: Solids and Liquids

Enthalpy is defined by $h=u+Pν$, where $P$ is the absolute pressure and $ν$ is the specific volume.  For a liquid and solid, the specific volume will be constant.  As a result,  the differential equation for enthalpy change of a liquid and solid is the following.

(Eq 4) $dh=du+νΔP+Pdν=du+νdP$

where $Pdν→0$

After integrating equation 4, the following equation will result.

(Eq 5) $Δh=Δu+νΔP≅c_{avg}ΔT+νΔP$ (kJ/kg)

Regarding solids, only the term $νΔP$ can be considered insignificant.  As a result, $Δh=Δu≅c_{avg}ΔT$.  On the other hand, for liquids, there are two common cases that can be encountered.  The first case is when there is a constant pressure during the process, or $ΔP=0$.  When this is the case, $Δh=Δu≅c_{avg}ΔT$.  The second case occurs when the temperature is constant during the process, or $ΔT=0$.  If this is the case than $Δh=νΔP$.

Finally, if we are trying to find the enthalpy of a compressed liquid, than the following equation is used.

(Eq 6) $h_{@P,T}≅h_{f@T}+ν_{f@T}(P-P_{sat@T})$

This equation is used when state 1 of a process starts as a saturated liquid.  Than as the process occurs the temperature remains the same until the process ends as a compressed liquid at state 2.

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