Question
Starting with the thermodynamic relationship G = H - TS, derive the following relationship: $\Delta\text{G}=-\text{T}\Delta\text{S}_\text{Total}$
✓
Answer
$\text{G}=\text{H}-\text{TS},$ $\text{G}_1=\text{H}_1-\text{TS}_1,$ $\text{G}_2=\text{H}_2-\text{TS}_2$$\text{G}_2-\text{G}_1=\text{H}_2-\text{H}_1-\text{T}(\text{S}_2-\text{S}_1)$
$\Delta\text{G}=\text{H}_2-\text{H}_1-\text{T}(\text{S}_2-\text{S}_1)$
$\Delta\text{G}=\Delta\text{H}-\text{T}\Delta\text{S}$
$\Delta\text{S}_\text{Total}=\Delta\text{S}_\text{sys}+\Delta\text{S}_\text{surr}$
$\Rightarrow\Delta\text{S}_\text{Total}=\Delta\text{S}_\text{sys}-\frac{\text{q}}{\text{T}}$
$\Delta\text{S}_\text{sys}=\frac{+\text{q}}{\text{T}},\Delta\text{S}_\text{surr}=\frac{-\text{q}}{\text{T}}$
$\Rightarrow\text{T}\Delta\text{S}_\text{Total}=\text{T}\Delta\text{S}_\text{sys}-\text{q}=\text{T}\Delta\text{S}_\text{sys}-\Delta\text{H}$
$\text{T}\Delta\text{S}_\text{Total}=-\Delta\text{G}$
$\Rightarrow\Delta\text{G}=-\text{T}\Delta\text{S}_\text{Total}$
$\Delta\text{G}=\text{H}_2-\text{H}_1-\text{T}(\text{S}_2-\text{S}_1)$
$\Delta\text{G}=\Delta\text{H}-\text{T}\Delta\text{S}$
$\Delta\text{S}_\text{Total}=\Delta\text{S}_\text{sys}+\Delta\text{S}_\text{surr}$
$\Rightarrow\Delta\text{S}_\text{Total}=\Delta\text{S}_\text{sys}-\frac{\text{q}}{\text{T}}$
$\Delta\text{S}_\text{sys}=\frac{+\text{q}}{\text{T}},\Delta\text{S}_\text{surr}=\frac{-\text{q}}{\text{T}}$
$\Rightarrow\text{T}\Delta\text{S}_\text{Total}=\text{T}\Delta\text{S}_\text{sys}-\text{q}=\text{T}\Delta\text{S}_\text{sys}-\Delta\text{H}$
$\text{T}\Delta\text{S}_\text{Total}=-\Delta\text{G}$
$\Rightarrow\Delta\text{G}=-\text{T}\Delta\text{S}_\text{Total}$
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