The gas phase decomposition of dimethylether follows first order kinetics `CH_3-O-CH_3(g)rarrCH_4(g)+H_2(g)+CO_((g))` The reaction is carried out in constant volume container at `500^@C` and has a half - life of 14.5 . Initially only dimethylether is present at a pressure of 0.40 atm . The total pressure of the system after 12 min is `x/100` atm . The value of x is [Given `10^(0.25)=1.778`]

Answer

To solve the problem, we will follow these steps:

### Step 1: Determine the rate constant (k)
Given the half-life of the reaction (t_half) is 14.5 minutes, we can calculate the rate constant (k) for a first-order reaction using the formula:

\[ k = \frac{0.693}{t_{half}} \]

Substituting the value:

\[ k = \frac{0.693}{14.5} \approx 0.0478 \, \text{min}^{-1} \]

### Step 2: Use the first-order kinetics equation
For a first-order reaction, the relationship between the initial pressure (P_0), the pressure at time t (P_t), and the rate constant (k) is given by:

\[ P_t = P_0 \cdot e^{-kt} \]

However, we can also use the logarithmic form:

\[ \log \left( \frac{P_0}{P_t} \right) = kt \cdot \frac{2.303}{1} \]

### Step 3: Calculate the pressure after 12 minutes
We know:
- Initial pressure, \( P_0 = 0.40 \, \text{atm} \)
- Time, \( t = 12 \, \text{min} \)

Substituting the values into the equation:

\[ \log \left( \frac{0.40}{P_t} \right) = 0.0478 \cdot 12 \cdot 2.303 \]

Calculating the right side:

\[ 0.0478 \cdot 12 \cdot 2.303 \approx 1.32 \]

Now, substituting back into the logarithmic equation:

\[ \log \left( \frac{0.40}{P_t} \right) = 1.32 \]

Converting from logarithmic form to exponential form:

\[ \frac{0.40}{P_t} = 10^{1.32} \]

Calculating \( 10^{1.32} \):

\[ 10^{1.32} \approx 20.9 \]

Thus,

\[ P_t = \frac{0.40}{20.9} \approx 0.0191 \, \text{atm} \]

### Step 4: Calculate the amount decomposed (A)
The amount of dimethyl ether that has decomposed after 12 minutes is:

\[ A = P_0 - P_t = 0.40 - 0.0191 \approx 0.3809 \, \text{atm} \]

### Step 5: Calculate the total pressure after 12 minutes
The total pressure (P_total) in the system is the sum of the remaining dimethyl ether and the products formed (CH4, H2, CO). Since the stoichiometry shows that for every mole of dimethyl ether that decomposes, one mole of each product is formed, we can express the total pressure as:

\[ P_{total} = P_t + 2A \]

Substituting the values:

\[ P_{total} = 0.0191 + 2(0.3809) = 0.0191 + 0.7618 \approx 0.7809 \, \text{atm} \]

### Step 6: Relate total pressure to x
We are given that the total pressure after 12 minutes is \( \frac{x}{100} \) atm. Thus:

\[ \frac{x}{100} = 0.7809 \]

Multiplying both sides by 100 to find x:

\[ x = 0.7809 \times 100 \approx 78.09 \]

Since we need x as an integer, we round it to:

\[ x \approx 78 \]

### Final Answer
Thus, the value of x is approximately **78**.

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