In the balanced equation : a Al + b NaOH + c H2O → d NaAlO2 + 3 H₂ (g)↑ the value of a b c and d are respectively

Question

Accepted Answer

To solve the equation $ a \text{ Al} + b \text{ NaOH} + c \text{ H}_2\text{O} \rightarrow d \text{ NaAlO}_2 + 3 \text{ H}_2 (g)↑ $, we need to balance the number of atoms on both sides of the equation.

### Step 1: Identify the Compounds
- **Reactants**: Aluminum (Al), Sodium Hydroxide (NaOH), Water (H₂O)
- **Products**: Sodium Aluminate (NaAlO₂) and Hydrogen gas (H₂)

### Step 2: Write the Unbalanced Equation
The unbalanced equation is:
$$ a \text{ Al} + b \text{ NaOH} + c \text{ H}_2\text{O} \rightarrow d \text{ NaAlO}_2 + 3 \text{ H}_2 $$

### Step 3: Balance the Equation
1. **Aluminum (Al)**: Each NaAlO₂ contains 1 Al, so if we have $ d $ NaAlO₂, we need $ d $ Al on the left side. Thus, $ a = d $.
2. **Sodium (Na)**: Each NaAlO₂ contains 1 Na, so we need $ d $ Na on the left side. Since NaOH contains 1 Na, we have $ b = d $.
3. **Oxygen (O)**: Each NaAlO₂ contains 2 O, so $ d $ NaAlO₂ contributes $ 2d $ O. NaOH contributes $ b $ O and H₂O contributes $ c $ O. Therefore, we have:
   $$ b + c = 2d $$
4. **Hydrogen (H)**: Each NaOH contributes 1 H and each H₂O contributes 2 H. The total H from the reactants is $ b + 2c $, and we have $ 3d $ H from the products (3 H₂). Thus:
   $$ b + 2c = 3d $$

### Step 4: Substitute and Solve
From our previous observations:
- Let $ d = 2 $ (this is a reasonable assumption to start with).
- Then $ a = d = 2 $ and $ b = d = 2 $.
- Substitute $ d = 2 $ into the equations:
  - From $ b + c = 2d $: $ 2 + c = 4 $ → $ c = 2 $
  - From $ b + 2c = 3d $: $ 2 + 2(2) = 6 $ → $ 6 = 6 $ (this checks out).

### Conclusion
Thus, the values of $ a, b, c, $ and $ d $ are:
- $ a = 2 $
- $ b = 2 $
- $ c = 2 $
- $ d = 2 $

The correct option is **2. 2, 2, 2, 2**.

### Why Other Options are Incorrect
1. **Option 1 (3, 2, 1, 3)**: This does not satisfy the balance of elements.
2. **Option 3 (3, 3, 3, 3)**: This also does not balance the equation correctly.
3. **Option 4 (2, 3, 2, 2)**: While $ a $ and $ d $ are correct, $ b $ does not satisfy the sodium balance.

Therefore, the only correct answer is **Option 2: 2, 2, 2, 2**.