Explanation:
To balance the photosynthesis equation using the algebraic method (abcd method), we'll follow these steps:
The photosynthesis equation is typically represented as:
\[ \text{6 CO}_2 + \text{6 H}_2\text{O} \rightarrow \text{C}_6\text{H}_{12}\text{O}_6 + \text{6 O}_2 \]
Here's how we can balance it using the abcd method:
1. **Assign variables to the coefficients**: Let's denote the coefficients of carbon dioxide (CO2), water (H2O), glucose (C6H12O6), and oxygen (O2) as \( a, b, c, \) and \( d \), respectively.
So, the equation becomes:
\[ a \text{ CO}_2 + b \text{ H}_2\text{O} \rightarrow c \text{ C}_6\text{H}_{12}\text{O}_6 + d \text{ O}_2 \]
2. **Count atoms for each element**: Count the atoms of each element on both sides of the equation.
**Carbon (C):**
- Left side: \( a \times 1 + c \times 6 = a + 6c \)
- Right side: \( c \times 6 = 6c \)
Equate and solve for \( a \):
\[ a + 6c = 6c \]
\[ a = 0 \]
So, \( a = 0 \).
**Hydrogen (H):**
- Left side: \( b \times 2 = 2b \)
- Right side: \( c \times 12 = 12c \)
Equate and solve for \( b \):
\[ 2b = 12c \]
\[ b = 6c \]
So, \( b = 6c \).
**Oxygen (O):**
- Left side: \( a \times 2 + b \times 1 = 2a + b \)
- Right side: \( d \times 2 \)
Equate and solve for \( d \):
\[ 2a + b = 2d \]
Substitute \( a = 0 \) and \( b = 6c \):
\[ 2 \times 0 + 6c = 2d \]
\[ 6c = 2d \]
\[ d = 3c \]
So, \( d = 3c \).
3. **Choose a value for \( c \)**: Choose a suitable integer value for \( c \) that makes \( b \) and \( d \) integers.
Let's choose \( c = 1 \):
- \( b = 6 \times 1 = 6 \)
- \( d = 3 \times 1 = 3 \)
4. **Write the balanced equation**: Substitute \( a = 0, b = 6, c = 1, d = 3 \) back into the equation:
\[ \text{6 CO}_2 + \text{6 H}_2\text{O} \rightarrow \text{C}_6\text{H}_{12}\text{O}_6 + \text{6 O}_2 \]
becomes
\[ 6 \text{ CO}_2 + 6 \text{ H}_2\text{O} \rightarrow \text{C}_6\text{H}_{12}\text{O}_6 + 6 \text{ O}_2 \]
Therefore, the balanced photosynthesis equation using the algebraic (abcd) method is:
\[ \boxed{6 \text{ CO}_2 + 6 \text{ H}_2\text{O} \rightarrow \text{C}_6\text{H}_{12}\text{O}_6 + 6 \text{ O}_2} \]
Each atom's count on both sides of the equation confirms that it is balanced:
- Carbon: \( 6 \times 1 = 6 \)
- Hydrogen: \( 6 \times 2 = 12 \)
- Oxygen: \( 6 \times 2 + 6 \times 1 = 12 + 6 = 18 \)
