Calculating ΔH Using Hess's Law: What Steps Are Involved?

[Eclair_de_XII]

Homework Statement

Given the data:
N₂(g) + O₂(g) → 2NO(g) ΔH = 180.7kJ
2NO(g) + O₂(g) → 2NO₂(g) ΔH = -113.1kJ
2N₂O(g) → 2N₂(g) + O₂(g) ΔH = -163.2kJ

use Hess's law to calculate ΔH for the reaction
N₂O(g) + NO₂(g) → 3NO(g)

Homework Equations

"Hess's law states that if a reaction is carried out in a series of steps, ΔH for the overall reaction equals the sum of the enthalpy changes for the individual steps."

The Attempt at a Solution

½(N₂(g) + O₂(g) → 2NO(g)) ΔH = 90.35kJ
½(2N₂O(g) → 2N₂(g) + O₂(g)) ΔH = -81.6kJ

3(N(g) + O(g) → NO(g)) ΔH = 271.05 kJ
½(2NO(g) + O₂(g) → 2NO₂(g)) ΔH = -56.55kJ
N₂(g) + O(g) → N₂O(g) ΔH = 81.6kJ

271.05 - (-56.55 + 81.6) = 246kJ

 

[Bystander]

3(N(g) + O(g) → NO(g)) ΔH = 271.05 kJ

Product, good.

N2(g) + O(g) → N2O(g) ΔH = 81.6kJ

One reactant, good.

½(2NO(g) + O2(g) → 2NO2(g)) ΔH = -56.55kJ

You omitted a step with this reactant.

 

[Eclair_de_XII]

Did I forget to distribute the half?

NO(g) + O(g) → NO₂(g) ΔH = -56.55kJ

 

[Bystander]

You forgot to "synthesize" the NO as far as getting a heat of formation for NO₂.

 

[Eclair_de_XII]

So -56.55 kJ isn't the heat of formation for nitrogen dioxide?

 

[Bystander]

It's the heat of formation of NO₂ from NO and oxygen, but not from nitrogen and oxygen, which would be the heat of formation as you've chosen to solve the problem.

 

[Chestermiller]

You need to find the coefficients of a linear combination of the three reactions that gives you the desired reaction. The third reaction is the only one that contains N2O, so it's coefficient must be 1/2. The second reaction is the only one that contains NO2, but it on the right hand side, so it's coefficient must be -1/2. To make good on these, the coefficient of the first reaction must be 1.

Chet

 

[Eclair_de_XII]

It's the heat of formation of NO₂ from NO and oxygen, but not from nitrogen and oxygen, which would be the heat of formation as you've chosen to solve the problem.

So is it (-113.1 - 180.7) = -293.8kJ?

You need to find the coefficients of a linear combination of the three reactions that gives you the desired reaction. The third reaction is the only one that contains N2O, so it's coefficient must be 1/2. The second reaction is the only one that contains NO2, but it on the right hand side, so it's coefficient must be -1/2. To make good on these, the coefficient of the first reaction must be 1.

I don't understand. I just added the coefficients so as to match the number of each gas in the desired equation. What are you talking about?

 

[Chestermiller]

So is it (-113.1 - 180.7) = -293.8kJ?
I don't understand. I just added the coefficients so as to match the number of each gas in the desired equation. What are you talking about?

$\frac{1}{2}(-163.2)-\frac{1}{2}(-113.1)+1(180.7)=155.65$

Here's a check on my answer:

Heat of formation of NO = 90.25

Heat of formation of NO2 = 33.18

Heat of formation of N2O = 82.05

3(90.25) - 33.18 - 82.05 = 155.52

How does that grab ya?

Chet

 

[Eclair_de_XII]

$\frac{1}{2}(-163.2)-\frac{1}{2}(-113.1)+1(180.7)=155.65$

I don't understand why you divided both the 2NO₂ and 2N₂O by two, whereas you left the 2NO untouched. I checked with a tutor and he came up with the same thing. So why aren't we multiplying the heat of formation of 2NO by 3/2?

 

[Chestermiller]

I don't understand why you divided both the 2NO₂ and 2N₂O by two, whereas you left the 2NO untouched. I checked with a tutor and he came up with the same thing. So why aren't we multiplying the heat of formation of 2NO by 3/2?

Because N2 and O2 are not involved in the reaction for which you are trying to find the heat of reaction.

Suppose you had 3 algebraic equations involving the 5 unknowns a, b, c, d, and e. The three equations are:

a + b = 2c

2c + b = 2d

2e = 2a +b

You are asked to form a linear combination of these three equations by multiplying each equation by a constant and then adding the resulting (new) three relationships together to obtain:

e + d = 3c

Are you able to figure out what the three constants would have to be in order for you to accomplish this?

Chet

 

[Eclair_de_XII]

Let's see:

½3(a + b) = ½3(2c)
½3a + ½3b = 3c
The constant is ½3.

½(2c + b) = ½(2d)
c + ½b = d

½(2e) = ½(2a + b)
e = a + ½b
The constant for the latter two is ½.

(a + ½b) + (c + ½b) = (½3a + ½3b)
a + b + c = (½3a + ½3b)
-½a - ½b + c = 0

 

[Chestermiller]

Let's see:

½3(a + b) = ½3(2c)
½3a + ½3b = 3c
The constant is ½3.

½(2c + b) = ½(2d)
c + ½b = d

½(2e) = ½(2a + b)
e = a + ½b
The constant for the latter two is ½.

(a + ½b) + (c + ½b) = (½3a + ½3b)
a + b + c = (½3a + ½3b)
-½a - ½b + c = 0

Well, that correctly gives a/2 + b/2 = c, which would give the heat of formation of NO (parameter c). Of course, that could have been obtained just you multiplying the first equation by 1/2.

But, the problem statement asks for the heat of the reaction e + d = 3c.

What do you get if you multiply the first equation by 1, the second equation by -1/2, and the third equation by +1/2 and add them together?

Chet

 

[Eclair_de_XII]

a + b = 2c

-½(2c + b) = -½(2d)
-c - ½b = -d
c + ½b = d

½2e = ½(2a + b)
e = a + ½b

(a + ½b) + (c + ½b) + a + b = 2a + 2b + c

I'm still having trouble with where you're going with this.

 

[Chestermiller]

a + b = 2c

-½(2c + b) = -½(2d)
-c - ½b = -d
c + ½b = d

½2e = ½(2a + b)
e = a + ½b

(a + ½b) + (c + ½b) + a + b = 2a + 2b + c

I'm still having trouble with where you're going with this.

a + b = 2c
-c - ½b = -d
e = a + ½b

Adding your 3 equations together:

(a + b) - (c + ½b) + e = 2c - d + (a + ½b)

Combining terms:

e + d = 3c

Chet

 

[Eclair_de_XII]

Oh, so I had to insert both sides of each equation into the final one?

 

[Chestermiller]

Oh, so I had to insert both sides of each equation into the final one?

Yes. That's what we have done. And note that the operations we carried out on the three equations are exactly the same as the operations we should perform on the three heats of reaction to arrive at the heat of the desired reaction.

Chet

 

[Eclair_de_XII]

Okay, thanks.

I appreciate the help.