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Suppose for a reaction: $$\ce{$a$A + $b$B -> $c$C + $d$D }$$

we know the $\Delta U$ as $\mathrm dU$ and all the components of the reaction are gas.

Then to calculate the $\Delta H$ at a temperature $T$ we use the expression $\mathrm dH=\mathrm dU+\left(c+d-a-b\right)RT$.

Then while using this expression did we assume the temperature to remain constant as it wouldn't really be constant as the actual expression for $\mathrm dH$ is: $\mathrm dH=\mathrm dU+\left(pV\right)$ and so we can write $\mathrm dH=\mathrm dU+\mathrm d(nRT)$ and for $\mathrm dH=\mathrm dU+\left(c+d-a-b\right)RT$, $T$ has to be constant?

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Yes. By saying that $\Delta (pV) = \Delta (nRT) = RT\Delta n$ you have made the assumption that $T$ is constant. When you take anything out of a $\Delta$ sign, you are saying that the initial and final values are the same. Otherwise you would have to write $\Delta (nRT) = R \Delta (nT) = R(n_2T_2 - n_1T_1)$ - this only simplifies to $RT(n_2 - n_1)$ if you assume $T_1 = T_2 = T$.


By the way, I want to be picky about the formulae that you are using. Look at $$\mathrm{d}H = \mathrm{d}U + (c + d - a - b)RT$$

The problem with this is that $\mathrm{d}H$ and $\mathrm{d}U$ are infinitesimals, but $(c + d - a - b)RT$ is a finite quantity. Do not ever add or subtract these two in the same equation. It is like trying to add a mass to a length - what is $30 \mathrm{~kg} + 25 \mathrm{~cm}$? It doesn't make sense.

Same thing applies for $\mathrm{d}H = \mathrm{d}U + (pV)$.

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  • $\begingroup$ In addition, the equation $dH = dU + d(pV)$ already assumes that temperature is constant before anything involving $\Delta$ terms. The fundamental equation is $dU = T dS - p dV$ or if you prefer $dH = T dS + V dP$ . The differential entropy $dS$ of an ideal gas depends on both temperature and volume. To neglect that term requires assuming $T$ is constant. $\endgroup$ – Curt F. Oct 2 '15 at 15:46
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When the heat of reaction is referred to, it means the enthalpy of the pure products (in stoichiometric proportions) minus the enthalpy or the pure reactants (in corresponding stoichiometric proportions), all at the same temperature (usually 25C) and pressure (usually 1 atm.).

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