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At $1100K$, solid $SiO_{2}$ has values of enthalpy and entropy of $−856.84[kJ/mol]$ and 124.51 [J/mol K], respectively. At $2500K$, liquid $SiO_{2}$ has values of enthalpy and entropy of $−738.44[kJ/mol]$ and $191.94[J/molK]$, respectively. The heat capacities of the solid and liquid phases are given by: $(C_{p})_{SiO_{2}}=53.466+0.02706T−1.27×10_{−5}T_{2}+2.19×10_{−9}T_{3}[J/(molK)](C_{p})_{SiO_{2}}=85.772[J/(molK)]. $ An alternative criteria for chemical equilibrium between two phases $α$ and $β$ of pure component " $i$ " can be written as: $(TG_{i} )_{α}=(TG_{i} )_{β}$ 1. Using calculus of thermodynamic equations first show that the partial derivative of this function with respect to temperature at constant pressure is given by $[∂T∂(TG ) ]_{P}=−T_{2}H_{i} .$ 2. Using only the data given, determine the temperature of the phase transition between solid and liquid. 3. Calculate the enthalpy of fusion.

1) To solve this problem, we'll need to apply the calculus of thermodynamic equations and use the given data. Let's go step by step.

First, we need to differentiate the alternative criteria for chemical equilibrium with respect to temperature at constant pressure:

Applying the chain rule:

Simplifying:

Dividing both sides by (dG_i / dT|P): ? = ?1 This implies that ? = ?1, which means the criterion for chemical equilibrium between the solid and liquid phases is satisfied when ? = ?1.