2.17 Internal energy
The conservation of energy Eq. (2.51 ) introduces the internal energy using the speciﬁc quantity , measured in in SI units. It represents the total molecular energy consisting of: kinetic energy associated with temperature; potential energy due to particle forces, both within particles as chemical bonds, and between particles, e.g. van der Waals forces.
The energies at diﬀerent scales can be summarised as:
 bulk — kinetic due to bulk motion, potential due to forces and ;
 molecular — kinetic characterised by , potential due to bonds.
To understand how energy is transferred between these scales, we can derive an equation for internal energy from Eq. (2.51 ), by cancelling terms in mechanical energy formed by taking the inner product of Eq. (2.19 ) with .

(2.56) 
The identity is the key element of the analysis. The term in Eq. (2.56 ) must represent the contribution of mechanical power to the internal energy, i.e. passing from bulk to molecular scale. Conversely, the term must contribute to mechanical energy.
Substituting from Eq. (2.41 ), Eq. (2.56) becomes

(2.57) 
If we substitute the Newtonian model from Eq. (2.41 )

(2.58) 
Double inner product of two tensors
The double inner product of two tensors, denoted by “”, is introduced in Eq. (2.56). It produces a scalar which is evaluated as the sum of the 9 products of the tensor components, eg:

(2.59) 