Answer :
To balance the chemical equation [tex]\( \square C_8H_{18} + \square O_2 \rightarrow \square CO_2 + \square H_2O \)[/tex], we need to ensure the number of atoms of each element is the same on both sides of the equation.
Let's denote the coefficients as follows:
- [tex]\( a C_8H_{18} \)[/tex]
- [tex]\( b O_2 \)[/tex]
- [tex]\( c CO_2 \)[/tex]
- [tex]\( d H_2O \)[/tex]
So the unbalanced chemical equation can be written as:
[tex]\[ a C_8H_{18} + b O_2 \rightarrow c CO_2 + d H_2O \][/tex]
### Step 1: Balance Carbon (C) Atoms
In [tex]\( C_8H_{18} \)[/tex], there are 8 carbon atoms. Each [tex]\( CO_2 \)[/tex] molecule has 1 carbon atom. Therefore, we need 8 [tex]\( CO_2 \)[/tex] molecules to balance the carbon atoms:
[tex]\[ a = 1 \quad \text{and} \quad c = 8 \][/tex]
### Step 2: Balance Hydrogen (H) Atoms
In [tex]\( C_8H_{18} \)[/tex], there are 18 hydrogen atoms. Each [tex]\( H_2O \)[/tex] molecule has 2 hydrogen atoms. Therefore, we need [tex]\( \frac{18}{2} = 9 \)[/tex] molecules of [tex]\( H_2O \)[/tex] to balance the hydrogen atoms:
[tex]\[ a = 1 \quad \text{and} \quad d = 9 \][/tex]
### Step 3: Balance Oxygen (O) Atoms
In the products, we have:
- [tex]\( 8 CO_2 \)[/tex] molecules contributing [tex]\( 8 \times 2 = 16 \)[/tex] oxygen atoms.
- [tex]\( 9 H_2O \)[/tex] molecules contributing [tex]\( 9 \times 1 = 9 \)[/tex] oxygen atoms.
So, the total number of oxygen atoms needed on the reactant side is:
[tex]\[ 16 + 9 = 25 \][/tex]
Each [tex]\( O_2 \)[/tex] molecule has 2 oxygen atoms. Therefore, we need [tex]\( \frac{25}{2} = 12.5 \)[/tex] molecules of [tex]\( O_2 \)[/tex]. To avoid fractional coefficients in the balanced chemical equation, we multiply all coefficients by 2:
[tex]\[ 1 \times 2 = 2 \quad (C_8H_{18}) \][/tex]
[tex]\[ 12.5 \times 2 = 25 \quad (O_2) \][/tex]
[tex]\[ 8 \times 2 = 16 \quad (CO_2) \][/tex]
[tex]\[ 9 \times 2 = 18 \quad (H_2O) \][/tex]
### Balanced Equation
The balanced equation, with all integer coefficients, is:
[tex]\[ 2 C_8H_{18} + 25 O_2 \rightarrow 16 CO_2 + 18 H_2O \][/tex]
### Final Answer
The coefficients that will balance the equation are:
[tex]\[ 2 \ C_8H_{18} + 25 \ O_2 \rightarrow 16 \ CO_2 + 18 \ H_2O \][/tex]
Let's denote the coefficients as follows:
- [tex]\( a C_8H_{18} \)[/tex]
- [tex]\( b O_2 \)[/tex]
- [tex]\( c CO_2 \)[/tex]
- [tex]\( d H_2O \)[/tex]
So the unbalanced chemical equation can be written as:
[tex]\[ a C_8H_{18} + b O_2 \rightarrow c CO_2 + d H_2O \][/tex]
### Step 1: Balance Carbon (C) Atoms
In [tex]\( C_8H_{18} \)[/tex], there are 8 carbon atoms. Each [tex]\( CO_2 \)[/tex] molecule has 1 carbon atom. Therefore, we need 8 [tex]\( CO_2 \)[/tex] molecules to balance the carbon atoms:
[tex]\[ a = 1 \quad \text{and} \quad c = 8 \][/tex]
### Step 2: Balance Hydrogen (H) Atoms
In [tex]\( C_8H_{18} \)[/tex], there are 18 hydrogen atoms. Each [tex]\( H_2O \)[/tex] molecule has 2 hydrogen atoms. Therefore, we need [tex]\( \frac{18}{2} = 9 \)[/tex] molecules of [tex]\( H_2O \)[/tex] to balance the hydrogen atoms:
[tex]\[ a = 1 \quad \text{and} \quad d = 9 \][/tex]
### Step 3: Balance Oxygen (O) Atoms
In the products, we have:
- [tex]\( 8 CO_2 \)[/tex] molecules contributing [tex]\( 8 \times 2 = 16 \)[/tex] oxygen atoms.
- [tex]\( 9 H_2O \)[/tex] molecules contributing [tex]\( 9 \times 1 = 9 \)[/tex] oxygen atoms.
So, the total number of oxygen atoms needed on the reactant side is:
[tex]\[ 16 + 9 = 25 \][/tex]
Each [tex]\( O_2 \)[/tex] molecule has 2 oxygen atoms. Therefore, we need [tex]\( \frac{25}{2} = 12.5 \)[/tex] molecules of [tex]\( O_2 \)[/tex]. To avoid fractional coefficients in the balanced chemical equation, we multiply all coefficients by 2:
[tex]\[ 1 \times 2 = 2 \quad (C_8H_{18}) \][/tex]
[tex]\[ 12.5 \times 2 = 25 \quad (O_2) \][/tex]
[tex]\[ 8 \times 2 = 16 \quad (CO_2) \][/tex]
[tex]\[ 9 \times 2 = 18 \quad (H_2O) \][/tex]
### Balanced Equation
The balanced equation, with all integer coefficients, is:
[tex]\[ 2 C_8H_{18} + 25 O_2 \rightarrow 16 CO_2 + 18 H_2O \][/tex]
### Final Answer
The coefficients that will balance the equation are:
[tex]\[ 2 \ C_8H_{18} + 25 \ O_2 \rightarrow 16 \ CO_2 + 18 \ H_2O \][/tex]
