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How do I solve problems where I'm asked for $\Delta H^\circ_f$ and given $\Delta H^\circ_f$ values for equations with the involved elements? For example:

Find the $\Delta H^\circ_f$ of the following reaction: $\ce{C + O_{2} -> CO _{2}}$, given:

$\ce{SrO + CO _{2} -> SrCO_{3}}$ ; $\Delta H^\circ_f = -234 \text{kJ}$

$\ce{2SrO -> 2Sr + O _{2}}$ ; $\Delta H^\circ_f = 1184 \text{kJ}$

$\ce{2SrCO _{3} -> 2Sr + 2C + 3O_{2}}$ ; $\Delta H^\circ_f H = 2440 \text{kJ}$

I know that I'm somehow supposed to manipulate the equations to make them "add up" (adding straight down) to the target equation, but I'm not sure how, exactly, to achieve this.

Also, is there a different and/or better method for doing these?

I've made some progress on my own, using this page. What I'm wondering now, is if elements are locked into compounds, can I split them up, as long as I keep them on the same side? In the example above, I've manipulated it to where I have $\ce{2CO _{2} + O _{2} -> 2C + 3O _{2}}$; am I allowed to split the $\ce{2CO _{2}}$ into $\ce{2C + 2O _{2}}$?

I've made it as far as getting to $\ce{2C + 2O _{2} -> 2CO _{2}}$. If you take these as molar ratios, it's the same as $\ce{C + O _{2} -> CO _{2}}$, which is the target equation. What do I have to do to the enthalpies to go from the first to the second?

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Hess's law isn't too difficult once you know what to look for. What may have caused you difficulty is 1.) either you are missing a reaction, or 2.) one of those listed reactions is incorrect. In order to properly use Hess's law, you need at least one reaction where $\ce{CO2}$ is present. – anosdsd Mar 13 '14 at 3:35
@Jun-GooKwak Am I approaching it the right way with the manipulating equations method? – evamvid Mar 13 '14 at 3:36
Not quite. You can either multiply those reactions by integers, or use the opposite reaction. Essentially you are trying to derive the equation that you have above by canceling out common elements on both sides of the products and reactants. In addition, you are keeping track of the enthalpies and find the sum. However, I do not see any reaction with carbon dioxide as a product or reactant. – anosdsd Mar 13 '14 at 3:37
@Jun-GooKwak You're right; on the problems sheet I got the question from, it's actually a $CO _{2}$ in the first one, not an $O _{2}$. – evamvid Mar 13 '14 at 3:41
Ah, excellent. It's solvable now. I'll do this example for you and I think it will be quite clear soon. – anosdsd Mar 13 '14 at 3:43
up vote 3 down vote accepted

Find the $\Delta H^\circ_f$ of the following reaction

$$\ce{C + O2 -> CO2}$$

We have our list of reactions with known enthalpies:

$\ce{SrO + CO2 -> SrCO3} $ $\Delta H^\circ_f = −234 \text{kJ}$

$\ce{2SrO -> 2Sr + O2}$ $\Delta H^\circ_f = 1184 \text{kJ} $

$\ce{2SrCO3 -> 2Sr + 2C + 3O2}$ $\Delta H^\circ_f = 2440 \text{kJ} $

Hess's Law takes advantage of the fact that enthalpy is a state function, looking at our reactions, we need to find a way to derive what we want:

Carbon dioxide is on the reactants side, however, we want it on the products, so we can run that reaction reverse; however we need to change the sign of enthalpy if we do that:

$\ce{SrCO3 -> SrO + CO2 } $ $\Delta H^\circ_f = 234 \text{kJ}$

We need to get rid of $\ce{SrO}$, thus, we use our second equation:

$\ce{2SrO -> 2Sr + O2 }$ $\Delta H^\circ_f = 1184 \text{kJ} $

Since, there's only one strontium oxide in the first reaction, we divide by the integer 2 across the chemical equation inculding the enthalpy of formation:

$\ce{SrO -> Sr + 1/2O2 }$ $\Delta H^\circ_f = 592 \text{kJ} $

Now, with equation 3, we need carbon and oxygen on the reactants side, and need to remove $\ce{SrCO3}$ and also divide again by 2, since we only have one strontium.

$\ce{Sr + C + 3/2O2 -> SrCO3 }$ $\Delta H^\circ_f = -1220 \text{kJ} $

Now we add up all our reactions and enthalpies, and cancel out common elements on both sides:

$\ce{SrCO3 -> SrO + CO2 } $ $\Delta H^\circ_f = 234 \text{kJ}$

$\ce{SrO -> Sr + 1/2O2 }$ $\Delta H^\circ_f = 592 \text{kJ} $

$\ce{Sr + C + 3/2O2 -> SrCO3 }$ $\Delta H^\circ_f = -1220 \text{kJ} $

$$\ce{C + O2 -> CO2}$$

$$\Delta H^\circ_f 234 \text{kJ} + \Delta H^\circ_f 592 \text{kJ} + \Delta H^\circ_f -1220 \text{kJ} = -394 \text{kJ}$$

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