Balancing a chemical equation with step-by-step calculations on notebook paper.

The image shows a chemical equation that needs to be balanced:

$$
a \text{PCl}_5 + b \text{H}_2\text{O} \rightarrow c \text{H}_3\text{PO}_4 + d \text{HCl}
$$

We are given the following steps and values:

- $ a = 1 $
- $ c = a = 1 $
- $ d = 5a = 5 $
- $ b = \frac{1}{2}(3 + 5) = 4 $

Let’s go through this step-by-step to understand how the equation is balanced.

---

Step 1: Understand the reaction

This is a hydrolysis reaction of phosphorus pentachloride (PCl₅), where it reacts with water (H₂O) to produce phosphoric acid (H₃PO₄) and hydrochloric acid (HCl):

$$
\text{PCl}_5 + \text{H}_2\text{O} \rightarrow \text{H}_3\text{PO}_4 + \text{HCl}
$$

We need to balance the atoms on both sides.

---

Step 2: Assign variables

Let:
- $ a $ = coefficient of PCl₅
- $ b $ = coefficient of H₂O
- $ c $ = coefficient of H₃PO₄
- $ d $ = coefficient of HCl

So:
$$
a \text{PCl}_5 + b \text{H}_2\text{O} \rightarrow c \text{H}_3\text{PO}_4 + d \text{HCl}
$$

---

Step 3: Balance atoms

#### Phosphorus (P):
- Left: $ a $
- Right: $ c $
→ So, $ a = c $

#### Chlorine (Cl):
- Left: $ 5a $ (since PCl₅ has 5 Cl)
- Right: $ d $ (since HCl has 1 Cl)
→ So, $ 5a = d $

#### Oxygen (O):
- Left: $ b $ (from H₂O)
- Right: $ 4c $ (since H₃PO₄ has 4 O)
→ So, $ b = 4c $

#### Hydrogen (H):
- Left: $ 2b $ (from H₂O)
- Right: $ 3c + d $ (from H₃PO₄ and HCl)
→ So, $ 2b = 3c + d $

---

Step 4: Use substitutions

We start with $ a = 1 $ (as given).

Then:
- $ c = a = 1 $
- $ d = 5a = 5 $
- Now use hydrogen balance: $ 2b = 3c + d = 3(1) + 5 = 8 $
→ $ b = 4 $

Now check oxygen:
- Left: $ b = 4 $ → 4 oxygen atoms
- Right: $ 4c = 4(1) = 4 $ → matches

So all atoms are balanced.

---

Final Balanced Equation:

$$
\text{PCl}_5 + 4 \text{H}_2\text{O} \rightarrow \text{H}_3\text{PO}_4 + 5 \text{HCl}
$$

---

Verification:

| Atom | Left Side | Right Side |
|------|-----------|------------|
| P | 1 | 1 |
| Cl | 5 | 5 |
| H | 4×2 = 8 | 3 + 5 = 8 |
| O | 4 | 4 |

✔ All atoms balance!

---

Conclusion:

The solution in the image is correct. The balanced equation is:

$$
\boxed{\text{PCl}_5 + 4\text{H}_2\text{O} \rightarrow \text{H}_3\text{PO}_4 + 5\text{HCl}}
$$

And the reasoning uses atomic balance with substitution based on known stoichiometry.