On dissolving 2.0 g of metal in sulphuric acid ,4.51 g of the metal sulphate was formed . The specific heat of the metal is 0.057 cal `g^(-1).^(@)C^(-1)`. What is the valency of metal ?

Answer

To find the valency of the metal, we can follow these steps:

### Step 1: Define the Problem
We are given:
- Mass of metal (M) = 2.0 g
- Mass of metal sulfate formed (MSO4) = 4.51 g
- Specific heat of metal = 0.057 cal/g/°C

We need to find the valency (n factor) of the metal.

### Step 2: Write the Chemical Formula
When the metal reacts with sulfuric acid (H2SO4), it forms a metal sulfate. Assuming the metal has a valency of \( x \), the formula for the metal sulfate can be written as \( M_2SO_4 \) (since sulfate has a charge of -2, two metals are needed to balance it).

### Step 3: Calculate the Equivalent Mass of Metal Sulfate
The equivalent mass of the metal sulfate can be calculated using the formula:
\[ \text{Equivalent mass} = \frac{\text{mass of metal sulfate}}{\text{number of equivalents}} \]
The number of equivalents of metal sulfate is the same as the number of equivalents of the metal.

### Step 4: Set Up the Equation for Equivalents
Let the equivalent mass of the metal be \( E_{metal} \). The number of equivalents of the metal is given by:
\[ \text{Number of equivalents of metal} = \frac{2.0 \, \text{g}}{E_{metal}} \]
And for the metal sulfate:
\[ \text{Number of equivalents of metal sulfate} = \frac{4.51 \, \text{g}}{E_{metal} + 48} \]
Here, \( 48 \) is the equivalent mass of sulfate (molar mass of sulfate = 96 g/mol, n factor = 2).

### Step 5: Equate the Two Expressions
Set the two expressions for the number of equivalents equal to each other:
\[ \frac{2.0}{E_{metal}} = \frac{4.51}{E_{metal} + 48} \]

### Step 6: Cross Multiply and Solve for \( E_{metal} \)
Cross multiplying gives:
\[ 2.0(E_{metal} + 48) = 4.51 E_{metal} \]
Expanding and rearranging:
\[ 2.0 E_{metal} + 96 = 4.51 E_{metal} \]
\[ 96 = 4.51 E_{metal} - 2.0 E_{metal} \]
\[ 96 = 2.51 E_{metal} \]
\[ E_{metal} = \frac{96}{2.51} \approx 38.24 \, \text{g/mol} \]

### Step 7: Calculate Molar Mass Using Specific Heat
Using the specific heat to find the molar mass:
\[ \text{Specific heat} \times \text{Molar mass} = 6.4 \]
\[ 0.057 \times \text{Molar mass} = 6.4 \]
\[ \text{Molar mass} = \frac{6.4}{0.057} \approx 112.28 \, \text{g/mol} \]

### Step 8: Calculate the Valency
Using the relationship between equivalent mass and molar mass:
\[ E_{metal} = \frac{\text{Molar mass}}{n} \]
Rearranging gives:
\[ n = \frac{\text{Molar mass}}{E_{metal}} = \frac{112.28}{38.24} \approx 3 \]

### Conclusion
The valency of the metal is **3**.

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