Question

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"`?

Answer 1

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.

Ref:
https://cbc-wb01x.chemistry.ohio-state.edu/~woodward/ch121/ch5_enthalpy.htm

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

Answer 2

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 `DeltaH_"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("Products"))(T) = C_(P("Reactants"))(T)` 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.

• https://www.et.byu.edu/~rowley/ChEn273/Topics/Energy_Balances/Reacting_Systems/Heat_of_Rxn_at_other_Temp.htm
  (boxed equation)

• https://www.slideshare.net/mobile/mosamgpatel/temperature-effect-on-heat-of-reaction
  (Eq. 7)

---

We can write a thermodynamic cycle for that.

• State 1: Reactants at `"273 K"`
• State 2: Products at `"273 K"`
• State 3: Reactants at `"373 K"`
• State 4: Products at `"373 K"`

`"State 1" stackrel(DeltaH_("rxn","273 K")" ")(->) "State 2"`

`"State 3" stackrel(DeltaH_("rxn","373 K")" ")(->) "State 4"`

`"State 1" stackrel(int_(273)^(373) C_(P("reactants"))(T)dT" ")(->) "State 3"`

`"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:

`"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 `DeltaH_"rxn"` for the reaction at `"373 K"`:

`color(blue)(DeltaH_("rxn","373 K")) = -int_(273)^(373) C_(P("reactants"))(T)dT + DeltaH_("rxn,273 K") + int_(273)^(373) C_(P("products"))(T)dT`

`= color(blue)(barul(|stackrel(" ")(" "DeltaH_("rxn,273 K") + int_(273)^(373) DeltaC_(P,"rxn")(T)dT" ")|))`

[See here; this person quotes the same equation I just derived.](https://chemistry.stackexchange.com/questions/35756/calculating-enthalpy-changes-at-different-temperatures) The astute chemist would recognize that this is just Hess's Law.

In other words, we can calculate `DeltaH_"rxn"` at `"373 K"` by cooling the reactants from `"373 K"` to `"273 K"`, using `DeltaH_"rxn"` at `"273 K"`, and then heating the products back up to `"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("Reactants"))(T) ~~ C_(P("Products"))(T)`?
• `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("products"))(T)dT-int_(273)^(373) C_(P("reactants"))(T)dT`

`= int_(273)^(373) DeltaC_(P,"rxn")dT ~~ 0`

since the integral of `DeltaC_(P,"rxn") ~~ 0` is zero. Only then is `DeltaH_("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("Products")) ne C_(P("Reactants"))` in general, then we really do NOT have the same `DeltaH_"rxn"` at two different temperatures:

`C_(P("products"))int_(273)^(373) dT - C_(P("reactants"))int_(273)^(373) dT`

`= DeltaC_(P,"rxn") int_(273)^(373) dT`

`= color(red)(DeltaC_(P,"rxn") cdot DeltaT ne 0)` in general