An ideal gas at initial temperature 300 K is compressed adiabatically (`gamma = 1.4`) to `(1/16)^(th)` of its initial volume. The gas is then expanded isobarically to double its volume. Then final temperature of gas round to nearest integer is:

Answer

To solve the problem step by step, we will analyze the two processes the ideal gas undergoes: adiabatic compression and isobaric expansion.

### Step 1: Understand the Initial Conditions
The initial temperature \( T_i \) of the gas is given as 300 K. The initial volume \( V_i \) is denoted as \( V_0 \).

### Step 2: Adiabatic Compression
The gas is compressed adiabatically to \( \frac{1}{16} \) of its initial volume. Therefore, the final volume after compression \( V_f \) is:
\[ V_f = \frac{V_0}{16} \]

For an adiabatic process, we can use the relation:
\[ T_i V_i^{\gamma - 1} = T_f V_f^{\gamma - 1} \]
where \( \gamma = 1.4 \).

### Step 3: Substitute Known Values
Substituting the known values into the equation:
\[ T_i V_0^{0.4} = T_f \left(\frac{V_0}{16}\right)^{0.4} \]

### Step 4: Simplify the Equation
Rearranging the equation gives:
\[ T_f = T_i \left(\frac{V_0}{\frac{V_0}{16}}\right)^{0.4} \]
This simplifies to:
\[ T_f = T_i \cdot 16^{0.4} \]

### Step 5: Calculate \( 16^{0.4} \)
Calculating \( 16^{0.4} \):
\[ 16^{0.4} = (2^4)^{0.4} = 2^{4 \cdot 0.4} = 2^{1.6} \]
Using a calculator or approximation, we find:
\[ 2^{1.6} \approx 3.17 \]

### Step 6: Calculate Final Temperature After Adiabatic Compression
Now substituting \( T_i = 300 \) K:
\[ T_f = 300 \cdot 3.17 \approx 951 \text{ K} \]

### Step 7: Isobaric Expansion
Next, the gas is expanded isobarically to double its volume. The new volume \( V_{new} \) after expansion is:
\[ V_{new} = 2 \cdot \left(\frac{V_0}{16}\right) = \frac{V_0}{8} \]

### Step 8: Use the Ideal Gas Law for Isobaric Process
For an isobaric process, the temperature is directly proportional to the volume:
\[ \frac{T_{new}}{T_f} = \frac{V_{new}}{V_f} \]
Substituting the known values:
\[ \frac{T_{new}}{951} = \frac{\frac{V_0}{8}}{\frac{V_0}{16}} = 2 \]
Thus,
\[ T_{new} = 2 \cdot 951 \approx 1902 \text{ K} \]

### Step 9: Round to Nearest Integer
Finally, rounding 1902 K to the nearest integer gives:
\[ T_{final} \approx 1902 \text{ K} \]

### Final Answer
The final temperature of the gas, rounded to the nearest integer, is **1902 K**.

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