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Nernst stated that the entropy change for each chemical or physical transition between condensed phases, at temperatures very close to the absolute zero, is equal to zero.
\[ \lim_{T\to 0} \Delta S = 0 \]
This statement is also referred to as the Nernst heat theorem. The statement of Nernst was simplified by Planck. He stated that not only the entropy change for processes but also the actual entropy of each condensed substance equals zero if the temperature approaches absolute zero.

3rd LoT states that, when the temperature of a pure crystalline substance approaches absolute 0K, its entropy approaches zero. \[ \boxed{ T \rightarrow 0 \Rightarrow S \rightarrow 0 } \]

Let's consider an example, pure orthorhombic A transform to stable monoclinic A at a temperature T K. \[ \textrm{As we know, } \Delta S^{0}_{Cycle} = 0 \] \[ \Rightarrow \Delta S^{0}_{I} + \Delta S^{0}_{II} + \Delta S^{0}_{III} - \Delta S^{0}_{trans} = 0 \] \[ \Rightarrow \Delta S^{0}_{trans} = \Delta S^{0}_{I} + \Delta S^{0}_{II} + \Delta S^{0}_{III} \] \[ \textrm{And as entropy change = 0 at T=0 K , } \Rightarrow \Delta S^{0}_{II} = 0 \]

Given \( S^{0}_{\textrm{monoclinic S, 0 }\rightarrow 400K} = 36.2 JK^{-1}\)

And, \( S^{0}_{\textrm{rhombodedral S, 0 }\rightarrow 400K}= 38.6 JK^{-1}\)

Answer : 2.4 \( JK^{-1} \)

\( \textrm{As , } \Delta S^{0}_{trans} = \Delta S^{0}_{I} + \Delta S^{0}_{II} +\Delta S^{0}_{III} \)

\( \Rightarrow \Delta S^{0}_{trans} = -36.2 + 0 + 38.6 = 2.4 JK^{-1}\)

1) If \( z = f(x,y) \)

\( dz = \Big(\frac{\partial z}{\partial x}\Big)_{y}dx + \Big(\frac{\partial z}{\partial y}\Big)_{x}dy \) \( \Rightarrow dz = Mdx + Ndy \)

So, If z is an exact differential \[ \Bigg(\frac{\partial M}{\partial y}\Bigg)_{x} = \Bigg(\frac{\partial N}{\partial x}\Bigg)_{y} \] 2) If \( z = f(x,y) \) \[ \Bigg(\frac{\partial z}{\partial x}\Bigg)_{y} . \Bigg(\frac{\partial x}{\partial y}\Bigg)_{z} . \Bigg(\frac{\partial y}{\partial z}\Bigg)_{x} = -1 \]

1) From TDS => TdS = dU + PdV

=> dU = TdS - PdV

\[ \Bigg(\frac{\partial T}{\partial V}\Bigg)_{S} = - \Bigg(\frac{\partial P}{\partial S}\Bigg)_{V} \]

2) From TDS => TdS = dH - VdP=> dH = TdS + VdP

\[ \Bigg(\frac{\partial T}{\partial P}\Bigg)_{S} = \Bigg(\frac{\partial V}{\partial S}\Bigg)_{P} \]

3) G = H - TS=> dG = dH - TdS - SdT = VdP - SdT

\[ \Bigg(\frac{\partial V}{\partial T}\Bigg)_{P} = - \Bigg(\frac{\partial S}{\partial P}\Bigg)_{T} \]

4) F = U - TS=> dF = dU - TdS - SdT = -PdV - SdT

\[ \Bigg(\frac{\partial P}{\partial T}\Bigg)_{V} = \Bigg(\frac{\partial S}{\partial V}\Bigg)_{T} \]