A mono atomic gas initially at `27^(@)C` is compressed adiabatically to one eighth of its original volume. The temperature after compression will be

Options

A. `10^(@)C`
B. `887^(@)C`
C. `927^(@)C`
D. `144^(@)C`

Explanation

To find the final temperature of a monoatomic gas after it has been compressed adiabatically to one-eighth of its original volume, we can use the adiabatic relation for an ideal gas. Here’s a step-by-step solution:

### Step 1: Convert the initial temperature to Kelvin
The initial temperature \( T_1 \) is given as \( 27^\circ C \). To convert this to Kelvin:
\[ T_1 = 27 + 273 = 300 \, K \]

### Step 2: Identify the initial and final volumes
Let the initial volume be \( V_1 \). According to the problem, the final volume \( V_2 \) after compression is:
\[ V_2 = \frac{V_1}{8} \]

### Step 3: Use the adiabatic relation
For an adiabatic process involving an ideal gas, the following relation holds:
\[ T_1 V_1^{\gamma - 1} = T_2 V_2^{\gamma - 1} \]
where \( \gamma \) (gamma) is the heat capacity ratio. For a monoatomic gas, \( \gamma = \frac{5}{3} \).

### Step 4: Substitute the known values into the equation
Substituting the known values into the adiabatic relation:
\[ 300 \cdot V_1^{\frac{5}{3} - 1} = T_2 \cdot \left(\frac{V_1}{8}\right)^{\frac{5}{3} - 1} \]
This simplifies to:
\[ 300 \cdot V_1^{\frac{2}{3}} = T_2 \cdot \left(\frac{V_1}{8}\right)^{\frac{2}{3}} \]

### Step 5: Simplify the equation
Rearranging the equation gives:
\[ T_2 = 300 \cdot \frac{V_1^{\frac{2}{3}}}{\left(\frac{V_1}{8}\right)^{\frac{2}{3}}} \]
This can be further simplified:
\[ T_2 = 300 \cdot \frac{V_1^{\frac{2}{3}}}{\frac{V_1^{\frac{2}{3}}}{8^{\frac{2}{3}}}} = 300 \cdot 8^{\frac{2}{3}} \]

### Step 6: Calculate \( 8^{\frac{2}{3}} \)
Calculating \( 8^{\frac{2}{3}} \):
\[ 8^{\frac{2}{3}} = (2^3)^{\frac{2}{3}} = 2^2 = 4 \]

### Step 7: Final calculation of \( T_2 \)
Now substituting back:
\[ T_2 = 300 \cdot 4 = 1200 \, K \]

### Conclusion
The final temperature \( T_2 \) after the adiabatic compression is:
\[ \boxed{1200 \, K} \]