# What is the enthalpy of reaction at "373 K" if the heat capacities are the same at "273 K" and we know DeltaH_(rxn) at "273 K"?

Nov 5, 2017

A) -20kJ

#### Explanation:

Enthalpy of a reaction is NOT affected by the temperature. The temperature may affect the reaction rate , but the inherent amount of heat energy produced or consumed by a reaction remains the same no matter what the conditions of the reaction environment are.

https://chem.libretexts.org/Core/Physical_and_Theoretical_Chemistry/Thermodynamics/State_Functions/Enthalpy/Heat_of_Reaction

Nov 5, 2017

Due to the assumption of the question that ${C}_{P}$ is equal between both the products and reactants at any given temperature, ONLY THEN can we say that $\Delta {H}_{\text{rxn}}$ did not change in that temperature range.

Otherwise, there is a small difference. Remember, this is just Hess's Law...

DeltaH_("rxn","373 K") = DeltaH_("rxn,273 K") + cancel(int_(273)^(373) DeltaC_(P,"rxn")(T)dT)^(~~ 0)

if ${C}_{P \left(\text{Products"))(T) = C_(P("Reactants}\right)} \left(T\right)$ for both temperatures.

See here for further discussion:
https://chemistry.stackexchange.com/questions/35756/calculating-enthalpy-changes-at-different-temperatures

And see here for further references to back up this answer.

We can write a thermodynamic cycle for that.

• State 1: Reactants at $\text{273 K}$
• State 2: Products at $\text{273 K}$
• State 3: Reactants at $\text{373 K}$
• State 4: Products at $\text{373 K}$

$\text{State 1" stackrel(DeltaH_("rxn","273 K")" ")(->) "State 2}$

$\text{State 3" stackrel(DeltaH_("rxn","373 K")" ")(->) "State 4}$

$\text{State 1" stackrel(int_(273)^(373) C_(P("reactants"))(T)dT" ")(->) "State 3}$

$\text{State 2" stackrel(int_(273)^(373) C_(P("products"))(T)dT" ")(->) "State 4}$

We know the first one, and in principle can determine the last two, but do not know the second one. If we want to go from $3$ to $4$, we can pick the path as follows:

$\text{Reactants at 373 K" -> "Reactants at 273 K" -> "Products at 273 K" -> "Products at 373 K}$

This path is then going to result in calculating $\Delta {H}_{\text{rxn}}$ for the reaction at $\text{373 K}$:

$\textcolor{b l u e}{\Delta {H}_{\text{rxn","373 K")) = -int_(273)^(373) C_(P("reactants"))(T)dT + DeltaH_("rxn,273 K") + int_(273)^(373) C_(P("products}}} \left(T\right) \mathrm{dT}$

$= \textcolor{b l u e}{\overline{\underline{| \stackrel{\text{ ")(" "DeltaH_("rxn,273 K") + int_(273)^(373) DeltaC_(P,"rxn")(T)dT" }}{|}}}}$

See here; this person quotes the same equation I just derived. The astute chemist would recognize that this is just Hess's Law.

In other words, we can calculate $\Delta {H}_{\text{rxn}}$ at $\text{373 K}$ by cooling the reactants from $\text{373 K}$ to $\text{273 K}$, using $\Delta {H}_{\text{rxn}}$ at $\text{273 K}$, and then heating the products back up to $\text{373 K}$.

NOTE: It is ultimately not entirely clear what the question means by "the heat capacities are the same".

Are we saying

• ${C}_{P \left(\text{Reactants"))(T) ~~ C_(P("Products}\right)} \left(T\right)$?
• C_(P("Reactants/Products"))("273 K") ~~ C_(P("Reactants/Products"))("373 K")?

I interpret it is meaning that the products and reactants each have the same heat capacities at a particular temperature, and not that they are constant in the temperature range irrespective of what they are in relation to each other.

In that special case,

${\int}_{273}^{373} {C}_{P \left(\text{products"))(T)dT-int_(273)^(373) C_(P("reactants}\right)} \left(T\right) \mathrm{dT}$

$= {\int}_{273}^{373} \Delta {C}_{P , \text{rxn}} \mathrm{dT} \approx 0$

since the integral of $\Delta {C}_{P , \text{rxn}} \approx 0$ is zero. Only then is $\Delta {H}_{\text{rxn", "373 K") = DeltaH_("rxn", "273 K}}$.

If the question means that C_(P("Reactants"))("373 K") ~~ C_(P("Reactants"))("273 K") and that C_(P("Products"))("373 K") ~~ C_(P("Products"))("273 K"), but that ${C}_{P \left(\text{Products")) ne C_(P("Reactants}\right)}$ in general, then we really do NOT have the same $\Delta {H}_{\text{rxn}}$ at two different temperatures:

${C}_{P \left(\text{products"))int_(273)^(373) dT - C_(P("reactants}\right)} {\int}_{273}^{373} \mathrm{dT}$

$= \Delta {C}_{P , \text{rxn}} {\int}_{273}^{373} \mathrm{dT}$

$= \textcolor{red}{\Delta {C}_{P , \text{rxn}} \cdot \Delta T \ne 0}$ in general