A state function is a property of a system that is defined solely by state. In scientific terms, this is often described as a function in which the value is independent of the path taken. In simpler terms, we can say that a state function is a property where only the final and end states matter. The opposite of a state function would be a path function, which depends on the path taken.
Imagine that you were driving from Los Angeles to New York and were considering two routes. The first route involves driving directly to New York, only taking breaks for supplies and sleep. The second route involves stopping at all of the popular tourist locations, even if they're out of the way. Alongside the trip, you're going to stop by Nevada and Ohio to pick up two of your friends. If you want to conserve your gasoline, which route would you take?
Clearly, the first route would reduce gasoline usage. In this case, we can say that gasoline usage is a path function because it depends on the path taken from Los Angeles to New York.
Now consider the number of friends in your car. Does the amount of friends in your car depend on the path you take? Absolutely not, because you're a good friend and are going to pick them up regardless of the route you take. In this case, we can say that the number of friends in your car is a state function, because it only depends on where you are, not how you got there.
Similarly, you can imagine a coordinate grid with points A and B plotted on it. There are an infinite number of ways to go from point A to point B: one could draw a straight line, or do some sort of weird wave. For state functions, it doesn't matter how the points are connected but only that they are connected. For path functions, the way that the points are connected matters.
Hess's Law states that the change in enthalpy (`DeltaH_("rxn")`) of a reaction is a state function. If we were asked to calculate the enthalpy of reaction for the reaction
`(1) A + B rArr C + D`
but were only given the enthalpy of reaction for `(2) B + E rArr C` and `(3) A rArr E + D`, we could calculate the enthalpy of reaction for (1) through (2) and (3).
`(2) B + E rArr C : DeltaH_((2))`
`(3) A rArr E + D : DeltaH_((3))`
`(2)+(3)= A + B + E rArr E + C + D : DeltaH_((2+3))`
The E's cancel out on both sides of the reaction, leaving us with reaction (1)
`(1)=(2)+(3)= A + B rArr C + D : DeltaH_((2+3))=DeltaH_((1))`
Therefore, `DeltaH_((1))=DeltaH_((2))+DeltaH_((3))`
By applying Hess's Law, we can determine the enthalpy of any reaction from the enthalpies of other reactions. From this, we can put Hess's Law in simpler terms:
`DeltaH_("rxn")` is the same whether the reaction occurs in one step or multiple steps.
There are several rules we use when combining substituent reactions. We saw the first one above:
1. When two equations are added to form a new equation, the `DeltaH_("new equation")` is equal to the sum of the `DeltaH` from the original two equations.
2. When flipping an equation, `DeltaH_("flipped")=-DeltaH`
`A+B rArr C : DeltaH`
`C rArr A + B : -DeltaH`
3. When multiplying an equation by a number, the `DeltaH` is multiplied by the same number.
`A+B rArr C : DeltaH`
`2A+2B rArr 2C : 2DeltaH`
With these rules, we can fully apply Hess's Law. When solving problems using Hess's Law, here are the steps you should follow:
1. Balance equations.
2. Flip around equations to place the desired reactants on the reactants side and products on the products side.
3. Multiply equations to give the desired coefficients.
4. Add equations to cancel out undesired reactants/products.
It's problem time!
#1. What is the enthalpy of `2C_((s))+2O_(2(g)) rArr 2CO_(2(g))` when the enthalpy of
`CO_(2(g)) rArr C_((s))+O_(2(g))` is `((394 "kJ")/("mol"))`
Look back at the rules for combining reactions. We want to get from the equation that we know the enthalpy of to the equation that we're trying to find the enthalpy of.
`CO_(2(g)) rArr C_((s))+O_(2(g)) : ((394 "kJ")/("mol"))`
Since `C` and `O_2` are on the reactants side in the equation that we're trying to get to, flip the equation to get those two reactants on the reactants side. Note what this does to the enthalpy.
`C_((s))+O_(2(g)) rArr CO_(2(g)) : ((-394 "kJ")/("mol"))` :
If we multiply this equation by 2, we have the equation we're trying to get to. Pay attention to the enthalpy.
`2C_((s))+2O_(2(g)) rArr 2CO_(2(g)) : ((-788 "kJ")/("mol"))`
Answer: `((-788 "kJ")/("mol"))`
#2. Find the enthalpy of the reaction `N_2H_(4(l))+CH_4O_((l)) rArr CH_2O_((g))+N_(2(g))+3H_(2(g))` given:
`(1) 2NH_(3(g)) rArr N_2H_(4(l))+H_(2(g)) : DeltaH=23 kJ`
`(2) 2NH_(3(g)) rArr N_(2(g)) +3H_(2(g)) : DeltaH=57 kJ`
`(3) CH_2O_((g)) +H_(2(g)) rArr CH_4O_((l)) : DeltaH=81 kJ`
Solving Hess's Law problems are like putting together puzzles. There are a few general rules to get you started, but the biggest part is trial and error. Just like how with puzzles, you find the corner and edge pieces first, with Hess's Law problems, we move the reactants to their proper side first.
Step 1). In the equation we're trying to solve for, `N_2H_(4(l))` is on the reactants side. Likewise, `CH_4O_((l))` is on the reactants side. The first order of business is to flip the constituent equations to reflect this. Pay attention to the effects doing this has on the `DeltaH`.
`(1) N_2H_(4(l))+H_(2(g)) rArr 2NH_(3(g)) : DeltaH=-(23 kJ)`
`(2) 2NH_(3(g)) rArr N_(2(g)) +3H_(2(g)) : DeltaH=57 kJ`
`(3) CH_4O_((l)) rArr CH_2O_((g)) +H_(2(g)) : DeltaH=-(81 kJ)`
Step 2). Believe it or not, that's all it takes! Granted, there will be much longer Hess's Law problems you'll encounter, but the purpose of this one was to show how Hess's Law can be applied to multiple equations. Add `(1),(2), "and" (3)` together.
`N_2H_(4(l))+H_(2(g)) + 2NH_(3(g)) +CH_4O_((l)) rArr 2NH_(3(g))+N_(2(g)) +3H_(2(g))+ CH_2O_((g)) +H_(2(g))`
Cancel out the reactants that are present on both sides and you'll end up with.
`N_2H_(4(l))+CH_4O_((l)) rArr CH_2O_((g))+N_(2(g))+3H_(2(g))`
Which is the equation that we're trying to find the enthalpy of! The enthalpy of this equation will just be the sum of the enthalpy values for the modified constituent equations:
`DeltaH=-(23 kJ)+ 57 kJ+(-(81 kJ))`
Answer: `DeltaH_("rxn")=-56 kJ`
Hess's Law problems can be pretty tedious and long. With proper practice, you'll begin recognizing patterns that'll make solving the problems more straightfoward. My advice is to label your steps clearly in a way that someone else reading your work would be able to pickup your thought process immediately. There'll be more problems to do in the final review of this section.
1. A state function is a variable that only depends on state. In other words, it only depends on the initial and final states.
2. Hess's Law states that enthalpy is a state function. The enthalpy of an equation can be found by manipulating constituent equations.