Answer :
To find the overall reaction from the given chemical equations, we will sum the equations step-by-step while ensuring the substances on both the reactant and product sides balance out. Here are the given chemical equations:
1. [tex]\( 2 \, NO_2(g) \rightarrow 2 \, NO(g) + O_2(g) \)[/tex]
2. [tex]\( 2 \, NO(g) \rightarrow N_2(g) + O_2(g) \)[/tex]
3. [tex]\( N_2(g) + 2 \, O_2(g) \rightarrow N_2O_4(g) \)[/tex]
### Step-by-Step Solution:
1. First Equation:
[tex]\[ 2 \, NO_2(g) \rightarrow 2 \, NO(g) + O_2(g) \][/tex]
This means that 2 molecules of nitrogen dioxide decompose to produce 2 molecules of nitric oxide and 1 molecule of oxygen gas.
2. Second Equation:
[tex]\[ 2 \, NO(g) \rightarrow N_2(g) + O_2(g) \][/tex]
Here, 2 molecules of nitric oxide decompose to produce 1 molecule of nitrogen gas and 1 molecule of oxygen gas.
3. Combining the First Two Equations:
First, add the first and second equations:
[tex]\[ (2 \, NO_2(g) \rightarrow 2 \, NO(g) + O_2(g)) + (2 \, NO(g) \rightarrow N_2(g) + O_2(g)) \][/tex]
Combine the products and reactants and then cancel out the common species on both sides:
[tex]\[ 2 \, NO_2(g) + 2 \, NO(g) \rightarrow 2 \, NO(g) + O_2(g) + N_2(g) + O_2(g) \][/tex]
Cancel out [tex]\( 2 \, NO(g) \)[/tex] from both sides:
[tex]\[ 2 \, NO_2(g) \rightarrow N_2(g) + 2 \, O_2(g) \][/tex]
4. Third Equation:
[tex]\[ N_2(g) + 2 \, O_2(g) \rightarrow N_2O_4(g) \][/tex]
This equation shows 1 molecule of nitrogen gas reacting with 2 molecules of oxygen gas to form dinitrogen tetroxide.
5. Combining All Equations:
Now, combine the result from step 3 with the third equation:
[tex]\[ (2 \, NO_2(g) \rightarrow N_2(g) + 2 \, O_2(g)) + (N_2(g) + 2 \, O_2(g) \rightarrow N_2O_4(g)) \][/tex]
Combine them:
[tex]\[ 2 \, NO_2(g) \rightarrow N_2(g) + 2 \, O_2(g) + N_2(g) + 2 \, O_2(g) \rightarrow N_2O_4(g) \][/tex]
Cancel out [tex]\( N_2(g) \)[/tex] and [tex]\( 2 \, O_2(g) \)[/tex] from both sides:
[tex]\[ 2 \, NO_2(g) \rightarrow N_2O_4(g) \][/tex]
### Final Overall Reaction:
[tex]\[ 2 \, NO_2(g) \rightarrow N_2O_4(g) \][/tex]
Thus, the overall reaction obtained by adding the given equations is:
[tex]\[ 2 NO_2(g) \rightarrow N_2O_4(g) \][/tex]
1. [tex]\( 2 \, NO_2(g) \rightarrow 2 \, NO(g) + O_2(g) \)[/tex]
2. [tex]\( 2 \, NO(g) \rightarrow N_2(g) + O_2(g) \)[/tex]
3. [tex]\( N_2(g) + 2 \, O_2(g) \rightarrow N_2O_4(g) \)[/tex]
### Step-by-Step Solution:
1. First Equation:
[tex]\[ 2 \, NO_2(g) \rightarrow 2 \, NO(g) + O_2(g) \][/tex]
This means that 2 molecules of nitrogen dioxide decompose to produce 2 molecules of nitric oxide and 1 molecule of oxygen gas.
2. Second Equation:
[tex]\[ 2 \, NO(g) \rightarrow N_2(g) + O_2(g) \][/tex]
Here, 2 molecules of nitric oxide decompose to produce 1 molecule of nitrogen gas and 1 molecule of oxygen gas.
3. Combining the First Two Equations:
First, add the first and second equations:
[tex]\[ (2 \, NO_2(g) \rightarrow 2 \, NO(g) + O_2(g)) + (2 \, NO(g) \rightarrow N_2(g) + O_2(g)) \][/tex]
Combine the products and reactants and then cancel out the common species on both sides:
[tex]\[ 2 \, NO_2(g) + 2 \, NO(g) \rightarrow 2 \, NO(g) + O_2(g) + N_2(g) + O_2(g) \][/tex]
Cancel out [tex]\( 2 \, NO(g) \)[/tex] from both sides:
[tex]\[ 2 \, NO_2(g) \rightarrow N_2(g) + 2 \, O_2(g) \][/tex]
4. Third Equation:
[tex]\[ N_2(g) + 2 \, O_2(g) \rightarrow N_2O_4(g) \][/tex]
This equation shows 1 molecule of nitrogen gas reacting with 2 molecules of oxygen gas to form dinitrogen tetroxide.
5. Combining All Equations:
Now, combine the result from step 3 with the third equation:
[tex]\[ (2 \, NO_2(g) \rightarrow N_2(g) + 2 \, O_2(g)) + (N_2(g) + 2 \, O_2(g) \rightarrow N_2O_4(g)) \][/tex]
Combine them:
[tex]\[ 2 \, NO_2(g) \rightarrow N_2(g) + 2 \, O_2(g) + N_2(g) + 2 \, O_2(g) \rightarrow N_2O_4(g) \][/tex]
Cancel out [tex]\( N_2(g) \)[/tex] and [tex]\( 2 \, O_2(g) \)[/tex] from both sides:
[tex]\[ 2 \, NO_2(g) \rightarrow N_2O_4(g) \][/tex]
### Final Overall Reaction:
[tex]\[ 2 \, NO_2(g) \rightarrow N_2O_4(g) \][/tex]
Thus, the overall reaction obtained by adding the given equations is:
[tex]\[ 2 NO_2(g) \rightarrow N_2O_4(g) \][/tex]