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I understand how to calculate the change in enthalpy and entropy for a reaction in standard conditions, but is there a way to calculate these values at non-standard conditions?

Also, is there any way that a reaction with positive change in gibbs free energy will occur?

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Do you know how to calculate the change in enthalpy and entropy for a substance as you change the conditions (T, P, ...) in a thermodynamically reversible fashion? Now take the reactants at "standard" conditions, transform them to reactant(s) at the non-standard conditions, react them, and take the product(s) back to the standard condition. Compare with the reaction occurring at standard conditions. Figure out the enthalpy and entropy differences between reacting at 'standard' and 'non-standard' conditions.

As for your second question, ponder why some reactions do not go to completion.

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1) Yes.

The change of entalpy of one mole with cnange of temperature is known as $C_p$, a heat capacity at constant pressure. It is, unfortunately, a function of temperature, so one has to perform integration of

$\Delta H= \int\limits^{T_{final}}_{T_{standard}}{C_p(T)}dT$

AFAIK, $C_p$ is usually approximated as cubic polynom $C_p=aT^3+bT^2 +cT +d $, where $a,b,c,d$ are found in reference literature. However, the situation is further complicated by phase change. In case one calculates enthalpy of water at $120^oC$, he will have to perform integration both for liquid and gas phases in their respective regions of stability and then add heat of evaporation of water.

The change of entropy is calculated in similiar manner, using the fact that

$\Delta S= \int \frac {dQ} T$

during heating to different temperature

$\Delta S= \int\limits^{T_{final}}_{T_{standard}}\frac{C_p(T)}TdT $

After obtaining entalpies and entropies of reagents and products at different temperature, one can obtain $\Delta G$ of reaction at same temperature.

As you can see it requires a lot of math, so such excercises are almost universally hated.

2)Technically speaking, most reactions move to point of equilibrium, where concentraions of reagents and products are guareded by constant of equilibrium, like

$K=\frac{[A]^a[B]^b...[E]^e}{[W]^w[X]^x...[Z]^z}$

for reaction

$\ce{w W + x X + ... z Z = a A + b B + ... + e E}$

so, if one start from products, i.e. when

$[W]=[X]=...=[Z]=0$

a sligh amount of reagents will appear as equilibrium is achieved.

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