Sure, let's balance the chemical equation step-by-step:
The given equation is:
[tex]\[ C_5H_{12} + O_2 \rightarrow CO_2 + H_2O \][/tex]
To balance this equation, we'll use coefficients for each compound and solve for these coefficients. Let's denote them as follows:
- [tex]\( a \)[/tex]: coefficient for [tex]\( C_5H_{12} \)[/tex]
- [tex]\( b \)[/tex]: coefficient for [tex]\( O_2 \)[/tex]
- [tex]\( c \)[/tex]: coefficient for [tex]\( CO_2 \)[/tex]
- [tex]\( d \)[/tex]: coefficient for [tex]\( H_2O \)[/tex]
The unbalanced chemical equation is then:
[tex]\[ a \cdot C_5H_{12} + b \cdot O_2 \rightarrow c \cdot CO_2 + d \cdot H_2O \][/tex]
Next, we need to balance the equation by ensuring that the number of atoms for each element is the same on both the reactant and product sides:
### 1. Balance Carbon (C)
- There are [tex]\( 5a \)[/tex] carbon atoms in [tex]\( a \cdot C_5H_{12} \)[/tex].
- There are [tex]\( c \)[/tex] carbon atoms in [tex]\( c \cdot CO_2 \)[/tex].
Setting them equal gives us the equation:
[tex]\[ 5a = c \][/tex]
### 2. Balance Hydrogen (H)
- There are [tex]\( 12a \)[/tex] hydrogen atoms in [tex]\( a \cdot C_5H_{12} \)[/tex].
- There are [tex]\( 2d \)[/tex] hydrogen atoms in [tex]\( d \cdot H_2O \)[/tex].
Setting them equal gives us the equation:
[tex]\[ 12a = 2d \][/tex]
### 3. Balance Oxygen (O)
- There are [tex]\( 2b \)[/tex] oxygen atoms in [tex]\( b \cdot O_2 \)[/tex].
- There are [tex]\( 2c \)[/tex] oxygen atoms in [tex]\( c \cdot CO_2 \)[/tex].
- There are [tex]\( d \)[/tex] oxygen atoms in [tex]\( d \cdot H_2O \)[/tex].
Setting them equal gives us the equation:
[tex]\[ 2b = 2c + d \][/tex]
We now have the following system of equations:
1. [tex]\( 5a = c \)[/tex]
2. [tex]\( 12a = 2d \)[/tex]
3. [tex]\( 2b = 2c + d \)[/tex]
### Solving the System of Equations
From equation (1):
[tex]\[ c = 5a \][/tex]
From equation (2):
[tex]\[ 12a = 2d \][/tex]
[tex]\[ d = 6a \][/tex]
Substitute [tex]\( c \)[/tex] and [tex]\( d \)[/tex] into equation (3):
[tex]\[ 2b = 2(5a) + 6a \][/tex]
[tex]\[ 2b = 10a + 6a \][/tex]
[tex]\[ 2b = 16a \][/tex]
[tex]\[ b = 8a \][/tex]
### Final Coefficients
- [tex]\( a \)[/tex]: This can be any non-zero number for simplification purposes, we usually take [tex]\( a = 1 \)[/tex] for simplicity.
- [tex]\( b = 8a \rightarrow b = 8 \)[/tex] if [tex]\( a = 1 \)[/tex].
- [tex]\( c = 5a \rightarrow c = 5 \)[/tex] if [tex]\( a = 1 \)[/tex].
- [tex]\( d = 6a \rightarrow d = 6 \)[/tex] if [tex]\( a = 1 \)[/tex].
So the balanced chemical equation will be:
[tex]\[ 1 \cdot C_5H_{12} + 8 \cdot O_2 \rightarrow 5 \cdot CO_2 + 6 \cdot H_2O \][/tex]
Or more simply:
[tex]\[ C_5H_{12} + 8O_2 \rightarrow 5CO_2 + 6H_2O \][/tex]
This is the fully balanced chemical equation.
