To solve the problem, we will follow these steps:
### Step 1: Write the combustion reaction
The organic compound has the formula \( C_nH_{2n+2} \). When it combusts in excess oxygen, it produces carbon dioxide and water. The balanced equation for the combustion reaction is:
\[
C_nH_{2n+2} + (3n + 1/2) O_2 \rightarrow n CO_2 + (n + 1) H_2O
\]
### Step 2: Calculate the initial conditions
We know the initial volume of the reaction vessel is \( 22.41 \, \text{L} \), the initial pressure is \( 2 \, \text{atm} \), and the initial temperature is \( 273 \, \text{K} \).
Using the ideal gas law \( PV = nRT \), we can find the number of moles of gas initially present.
\[
n = \frac{PV}{RT} = \frac{(2 \, \text{atm})(22.41 \, \text{L})}{(0.0821 \, \text{L atm/(mol K)})(273 \, \text{K})}
\]
Calculating this gives:
\[
n \approx \frac{44.82}{22.414} \approx 2 \, \text{moles}
\]
### Step 3: Analyze the final conditions
After combustion, the final temperature is \( 546 \, \text{K} \) and the final pressure is \( 4.6 \, \text{atm} \).
Using the ideal gas law again for the final conditions:
\[
n' = \frac{PV}{RT} = \frac{(4.6 \, \text{atm})(22.41 \, \text{L})}{(0.0821 \, \text{L atm/(mol K)})(546 \, \text{K})}
\]
Calculating this gives:
\[
n' \approx \frac{103.606}{44.866} \approx 2.31 \, \text{moles}
\]
### Step 4: Determine the change in moles
The increase in moles due to combustion can be calculated as:
\[
\Delta n = n' - n = 2.31 - 2 = 0.31 \, \text{moles}
\]
### Step 5: Relate the increase in moles to \( n \)
From the balanced equation, we know that:
\[
\Delta n = (n + (n + 1)) - (1) = 2n + 1 - 1 = 2n
\]
Setting this equal to the increase in moles gives:
\[
2n = 0.31 \implies n = 0.155
\]
### Step 6: Calculate the molecular weight of the compound
The molecular weight of the compound \( C_nH_{2n+2} \) is:
\[
M = 12n + (2n + 2) = 14n + 2
\]
Substituting \( n = 4 \):
\[
M = 14(4) + 2 = 56 + 2 = 58 \, \text{g/mol}
\]
### Step 7: Determine the formula of the organic compound
Using \( n = 4 \):
\[
C_nH_{2n+2} = C_4H_{2(4)+2} = C_4H_{10}
\]
Thus, the formula of the organic compound is \( C_4H_{10} \).
### Final Answer
The formula of the organic compound is \( C_4H_{10} \).
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