**show: ΔH = q _{p}**

enthalpy, H, is defined as:

H = E + PV

"taking the derivative" [actually, should evalute the "differential"]

dH = dE + d(PV)

= dE + PdV + VdP [1]. For a constant pressure process, dP = 0; thus [1] becomes

dH = dE + PdV [2]. Work, W, is defined as:

W = - F * x

or dW = - F dx [3] where F = force and x= displacement.

Pressure, P, is defined as:

P = F / A

or F = P A [4] substituting [4] into [3],

dW = - P A dx = - P dV [5] substituting [5] into [2],

dH = dE - dW [6]. The 1

^{st}law of thermodynamics,

dE = dq + dW [7] substituting [7] into [6],

dH = dq + dW - dW

dH = dq

hence,

ΔH = q

_{p}.

**show ΔG < 0 for a spontaneous process.**

The 2

^{nd }law of thermodynamics states that for a spontaneous process,

ΔS _{universe }= Δ S_{system}+ Δ S_{surrounding}> 0[1]. As

q

_{system}= - q_{surrounding}and at constant pressure (see above),

q

_{system}= Δ Hthus

q

_{surrounding}= - Δ H.Dividing by T and using the definition of Δ S,

Δ S _{surrounding}= q_{surrounding}/ T = - Δ H / T[2]

substituting [2] into [1],

Δ S

_{universe}= Δ S_{system}+ - Δ H / T > 0multiplying by - T,

- T Δ S

_{universe}= - T Δ S_{system }+ Δ H < 0.Defining the change in Gibbs Free Energy as:

Δ G = - T Δ S

_{universe}substituting and rearranging,

Δ G = Δ H - T Δ S < 0 for a spontaneous process.