The reaction between ethylene C2H4 and hydrogen bromide HBr to form ethyl bromide C2H5Br is carried out in a continuous reactor. The product stream is analysed and found to contain 51.7 mol % C2H5Br and 17.3 mol % HBr, with the balance made up of C2H4 . The feed stream into the reactor contains only ethylene and hydrogen bromide. If the molar flow rate of the feed stream is 165 mol/s, the extent of reaction ξ is equal to?

To solve this problem, we need to calculate the extent of the reaction (denoted as \(\xi\)) for the reaction:

\[
\text{C}_2\text{H}_4 + \text{HBr} \rightarrow \text{C}_2\text{H}_5\text{Br}
\]

### Step 1: Let’s define variables.

- Molar flow rate of feed stream: \( F = 165 \, \text{mol/s} \)
- Composition of the product stream:
  - Ethyl bromide (\(\text{C}_2\text{H}_5\text{Br}\)): 51.7 mol%
  - Hydrogen bromide (\(\text{HBr}\)): 17.3 mol%
  - Ethylene (\(\text{C}_2\text{H}_4\)): \( 100 - 51.7 - 17.3 = 31.0 \, \text{mol\%} \)

Now, let’s assume the initial molar feed rates:
- Let the initial molar flow rates of \(\text{C}_2\text{H}_4\) and \(\text{HBr}\) be \( F_{\text{C}_2\text{H}_4}^0 = F_{\text{HBr}}^0 = \frac{F}{2} = 82.5 \, \text{mol/s} \), as the feed stream contains only these two reactants in equal molar amounts.

### Step 2: Analyze the product stream.

Let the extent of reaction be \(\xi\) (in mol/s).

- For \(\text{C}_2\text{H}_4\), the flow rate in the product stream is \( F_{\text{C}_2\text{H}_4} = F_{\text{C}_2\text{H}_4}^0 - \xi \).
- For \(\text{HBr}\), the flow rate in the product stream is \( F_{\text{HBr}} = F_{\text{HBr}}^0 - \xi \).
- For \(\text{C}_2\text{H}_5\text{Br}\), the flow rate in the product stream is \( F_{\text{C}_2\text{H}_5\text{Br}} = \xi \).

From the mole fractions given, the total flow rate in the product stream is:

\[
F_{\text{total}} = \frac{F_{\text{C}_2\text{H}_5\text{Br}}}{0.517} = \frac{\xi}{0.517}
\]

### Step 3: Relate the composition to the flow rates.

1. The mole fraction of \(\text{C}_2\text{H}_4\) is:

\[
\frac{F_{\text{C}_2\text{H}_4}}{F_{\text{total}}} = 0.310 \Rightarrow \frac{82.5 - \xi}{\frac{\xi}{0.517}} = 0.310
\]

2. The mole fraction of \(\text{HBr}\) is:

\[
\frac{F_{\text{HBr}}}{F_{\text{total}}} = 0.173 \Rightarrow \frac{82.5 - \xi}{\frac{\xi}{0.517}} = 0.173
\]

### Step 4: Solve for \(\xi\).

We’ll use the first relation to find \(\xi\). Let’s solve the equation:

\[
\frac{82.5 - \xi}{\frac{\xi}{0.517}} = 0.310
\]

\[
82.5 - \xi = 0.310 \cdot \frac{\xi}{0.517}
\]

Solving for \(\xi\).The extent of reaction (\(\xi\)) is approximately \(51.57 \, \text{mol/s}\).

Would you like further details on this calculation or any specific steps?

Here are 5 related questions you may want to explore:
1. How does the extent of reaction relate to the conversion in chemical reactions?
2. What other methods can be used to calculate the extent of reaction in different systems?
3. How would the extent of reaction change if the molar fractions of products were different?
4. How does a continuous reactor's performance compare to a batch reactor for this reaction?
5. What is the importance of knowing the extent of reaction in reactor design?

**Tip:** In chemical reaction engineering, mole balances and extent of reaction are crucial for scaling up processes to industrial levels.

The reaction between ethylene C2H4 and hydrogen bromide HBr to form ethyl bromide C2H5Br is carried out in a continuous reactor. The product stream is analysed and found to contain 51.7 mol % C2H5Br and 17.3 mol % HBr, with the balance made up of C2H4. The feed stream into the reactor contains only ethylene and hydrogen bromide. If the molar flow rate of the feed stream is 165 mol/s, the extent of reaction ξ is equal to?

Find out how to calculate the extent of reaction (ξ) in a continuous reactor with ethylene (C2H4) and hydrogen bromide (HBr) to form ethyl bromide (C2H5Br). This detailed guide explains stoichiometric principles, mole balances, and key formulas used in chemical reaction engineering, suitable for undergraduate students.

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