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Helmholtz Free Energy

F = U - TS, where F is Helmholtz free energy, U is internal energy, T is temperature, and S is entropy.
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The statement of the theorem

Let the thermodynamic state of a closed system be defined by the set of extensive variables X=(S,V)\mathbf{X} = (S, V). The internal energy UU is a state function, U=U(S,V)U = U(S, V). The fundamental thermodynamic relation is given by dU=TdS−PdVdU = T dS - P dV. The Helmholtz Free Energy FF is defined as the Legendre transform of UU with respect to S/TS/T, or equivalently, as a function of temperature TT and volume VV: F(T,V)=U(S,V)−TSF(T, V) = U(S, V) - T S. The differential change in FF is then derived by total differentiation: dF=d(U−TS)=dU−TdS−SdTdF = d(U - TS) = dU - T dS - S dT Substituting the fundamental relation dU=TdS−PdVdU = T dS - P dV: dF=(TdS−PdV)−TdS−SdTdF = (T dS - P dV) - T dS - S dT Simplifying yields the rigorous differential form: dF=−SdT−PdVdF = -S dT - P dV Furthermore, the partial derivatives define the system's response functions: \left(\frac{\partial F}{\partial T}\right)_{V} = -S \quad \text{and} \nn\left(\frac{\partial F}{\partial V}\right)_{T} = -P
Source: Wikipedia