1 kg of ethylene undergo polymerization completely, then the weight of polyethylene obtained is `10^x` gm. What is x ?

Explanation

To solve the problem, we need to determine the weight of polyethylene obtained from 1 kg of ethylene after complete polymerization. Let's break down the steps:

### Step 1: Understand the Monomer
The monomeric unit of polyethylene is ethylene, which has the molecular formula \(C_2H_4\). The molecular weight of ethylene can be calculated as follows:
- Carbon (C) has an atomic weight of approximately 12 g/mol.
- Hydrogen (H) has an atomic weight of approximately 1 g/mol.

Thus, the molecular weight of ethylene (\(C_2H_4\)) is:
\[ \text{Molecular weight of } C_2H_4 = (2 \times 12) + (4 \times 1) = 24 + 4 = 28 \text{ g/mol} \]

### Step 2: Calculate Moles of Ethylene
Given that we have 1 kg of ethylene, we convert this to grams:
\[ 1 \text{ kg} = 1000 \text{ g} \]
Now, we can calculate the number of moles of ethylene using the formula:
\[ \text{Moles of ethylene} = \frac{\text{Given mass}}{\text{Molecular weight}} = \frac{1000 \text{ g}}{28 \text{ g/mol}} \approx 35.71 \text{ moles} \]

### Step 3: Determine the Weight of Polyethylene
When ethylene polymerizes, it forms polyethylene. The polymerization process does not change the number of carbon atoms; it just links them together. Each mole of ethylene contributes to forming one mole of polyethylene.

Thus, the weight of polyethylene produced can be calculated as follows:
\[ \text{Weight of polyethylene} = \text{Moles of polyethylene} \times \text{Molecular weight of polyethylene} \]
Since polyethylene is made up of the same monomer (ethylene), its molecular weight can be approximated as:
\[ \text{Molecular weight of polyethylene} \approx 28 \text{ g/mol} \text{ (for simplicity, as it is a long chain)} \]
Therefore, the weight of polyethylene is:
\[ \text{Weight of polyethylene} = 35.71 \text{ moles} \times 28 \text{ g/mol} \approx 1000 \text{ g} \]

### Step 4: Express the Weight in Scientific Notation
The weight of polyethylene obtained is 1000 g. We can express this in scientific notation:
\[ 1000 \text{ g} = 10^3 \text{ g} \]

### Step 5: Determine the Value of x
From the expression \(10^x\), we can see that:
\[ x = 3 \]

### Final Answer
Thus, the value of \(x\) is:
\[ \boxed{3} \]

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