# Work Function: Helmholtz Free Energy

Work function is also called Helmholtz free energy. It is a state function measures the valuable work obtained from a closed thermodynamic system at a constant temperature. Under isothermal process (at constant temperature), the system exchanges heat with their surrounding.

Work function is defined as the internal energy minus the product of temperature and entropy. Mathematically it can be written as-

A = U − TS Equation:1

wheere, A is work function, U is internal energy, T and S are temperature and entropy respectively.

Let us consider a system which change from one state to other state at constant temperature. Work function (A) is a state function, so depends only on initial and final states.

Now we can write work function for two states as-

A_{1} = E_{1} − TS_{1}

Similarly,

A_{2} = E_{2} − TS_{2}

So, the change in work function

A_{2} − A_{1} = (E_{2} − TS_{2}) − (E_{2} − TS_{2})

or, ΔA = ΔE − TΔS

or, ΔA = ΔE − q (As ΔS = q/T)

We know that from 1st law of thermodynamics

ΔE = q − w

or, ΔE − q = − w

or, − ΔA = w

So, decrease in the work function in any process (not just the compressional (PV) work) at constant temperature gives the maximum work that can be obtained from the system.

The free energy function is defined as-

ΔF = H − TS

For finite change at constant temperature, we have-

ΔF = ΔH − TΔS

or, ΔF = ΔE + PΔV − TΔS (As ΔH = ΔE + PΔV)

or, ΔF = ΔA + PΔV (As ΔA = ΔE − TΔS)

or, ΔF = − w + PΔV

or, − ΔF = w − PΔV

Since, PΔV is the work done due to expansion against a constant pressure. So, the decrease in free energy (−ΔF) accompanying a process which occurs at constant temperature and pressure is the maximum work obtainable from the system other than PΔV. Hence, it is called as net work.

Net Work = − ΔF

This is of great importance as the cahnge in free energy is a measure of the net work which may be electrical, chemical or surface work.

## Variation of Work Function with Temperature and Volume

We know work function is represented as

A = E − TS

Differentiation of this equation gives

dA = dE – TdS – SdT

but dE = TdS – PdV (from final and second law of thermodynamics.

dA = TdS – PdV – TdS – SdT Equation:1

dA = – PdV – SdT Equation:2

At constant temperature,

SdT = 0

So, the equation:2 becomes-

(dA)_{T} = – PdV

or, (dA/dV)_{T} = – P

And at constant volume,

PdV = 0

So, the equation:2 becomes-

(dA)_{V} = – SdT

or, (dA/dT)_{V} = – S