# When can you express the Helmholtz energy as a sum of terms?

In both excess Gibbs/Helmholtz energy models the total Gibbs/Helmholtz energy is the sum of multiple contributions. For example, in the SAFT equation of state the total Helmholtz energy is

$$A = A^\text{KE} + A^\text{HS} + A^\text{disp} + A^\text{chain} + A^\text{assoc}\,.$$ Here $$\text{KE}$$, $$\text{HS}$$, $$\text{disp}$$, $$\text{chain}$$, and $$\text{assoc}$$ denote the kinetic energy, hard-sphere, dispersion, chain, and association contributions to the Helmholtz energy.

I am struggling to understand when you can write the Helmholtz energy as the sum of multiple terms. Presumably each of the Helmholtz energy contributions corresponds to a term in the Hamiltonian

$$H = \text{KE}(\mathbf{p}_1,\cdots, \mathbf{p}_N) + U^\text{HS}(\mathbf{q}_1,\cdots, \mathbf{q}_N)+ U^\text{disp}(\mathbf{q}_1,\cdots, \mathbf{q}_N)+ U^\text{chain}(\mathbf{q}_1,\cdots, \mathbf{q}_N)+ U^\text{assoc}(\mathbf{q}_1,\cdots, \mathbf{q}_N) \,.$$

Here $$H$$ is the Hamiltonian, $$\mathbf{p}_i$$ is the momentum of molecule $$i$$ and $$\mathbf{q}_i$$ is the position of molecule $$i$$. $$U^\text{x}$$ indicates the total pairwise energy of type $$\text{x}$$ between $$N$$ molecules. $$\text{KE}$$ is the total kinetic energy.

Is it an approximation or can you show from the Canonical partition function (see below) that each Helmholtz energy corresponds to a particular term in the Hamiltonian?

Note: The partition function for a continuous and classical system with N molecules is $$\mathcal{Z} = \frac{1}{h^{3N}N!} \int e^{-\beta H(p_1,\cdots,p_N,q_1,\cdots,q_N)}\text{d}^3\mathbf{p}_1\cdots\text{d}^3\mathbf{p}_N\text{d}^3\mathbf{q}_1\cdots\text{d}^3\mathbf{q}_N.$$ Here $$h$$ is Planck's constant and $$\beta$$ is the inverse temperature.

The partition function is related to the Helmholtz energy through $$A=-\frac{1}{\beta}\ln \mathcal{Z}$$.

The way you wrote the Hamiltonian, $$H = \text{KE}(\mathbf{p}_1,\cdots, \mathbf{p}_N) + U^\text{HS}(\mathbf{q}_1,\cdots, \mathbf{q}_N)+ U^\text{disp}(\mathbf{q}_1,\cdots, \mathbf{q}_N)+ U^\text{chain}(\mathbf{q}_1,\cdots, \mathbf{q}_N)+ U^\text{assoc}(\mathbf{q}_1,\cdots, \mathbf{q}_N) ,$$ would only allow expressing the Helmholtz free energy as a sum of an ideal gas contribution (originating from the kinetic energy term) and a configurational contribution from all the potential energy terms depending on the same degrees of freedom.