A closed vessel contains 0.1 mole of a monatomic ideal gas at 200k . If 0.05 mole of the same gas at 400 K is added to it , the final equilibrium temperature (in K ) of the gas in the vessel will be close to ...............

Answer

To find the final equilibrium temperature of the gas in the closed vessel after mixing two different amounts of the same gas at different temperatures, we can use the principle of conservation of energy. Here’s a step-by-step solution:

### Step 1: Identify the given values
- For the first gas:
- Number of moles, \( N_1 = 0.1 \) moles
- Initial temperature, \( T_1 = 200 \, K \)

- For the second gas:
- Number of moles, \( N_2 = 0.05 \) moles
- Initial temperature, \( T_2 = 400 \, K \)

### Step 2: Use the formula for final temperature
The final equilibrium temperature \( T \) can be calculated using the formula:
\[ T = \frac{N_1 T_1 + N_2 T_2}{N_1 + N_2} \]

### Step 3: Substitute the values into the formula
Substituting the known values into the equation:
\[ T = \frac{(0.1 \, \text{moles} \times 200 \, K) + (0.05 \, \text{moles} \times 400 \, K)}{0.1 \, \text{moles} + 0.05 \, \text{moles}} \]

### Step 4: Calculate the numerator
Calculating the numerator:
\[ 0.1 \times 200 = 20 \]
\[ 0.05 \times 400 = 20 \]
Adding these together:
\[ 20 + 20 = 40 \]

### Step 5: Calculate the denominator
Calculating the denominator:
\[ 0.1 + 0.05 = 0.15 \]

### Step 6: Calculate the final temperature
Now, substituting back into the equation:
\[ T = \frac{40}{0.15} \]
Calculating this gives:
\[ T = 266.67 \, K \]

### Final Answer
The final equilibrium temperature of the gas in the vessel will be approximately \( 266.67 \, K \).

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