A gas is compressed adiabatically till its pressure becomes 27 times its initial pressure. Calculate final temperature if initial temperature is`27^@ C` and value of `gamma` is 3/2.

Options

A. `300 K`
B. `600 K`
C. `900 K`
D. `1200 K`

Explanation

To solve the problem of finding the final temperature of a gas that is compressed adiabatically until its pressure becomes 27 times its initial pressure, we can follow these steps:

### Step 1: Convert 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 Given Values
- Initial pressure \( P_1 = P \) (let's denote it as \( P \))
- Final pressure \( P_2 = 27P \)
- Initial temperature \( T_1 = 300 \, K \)
- Adiabatic constant \( \gamma = \frac{3}{2} \)

### Step 3: Use the Adiabatic Relation
For an adiabatic process, the relationship between pressure and temperature is given by:
\[ \frac{P_1}{P_2} = \left(\frac{T_1}{T_2}\right)^{\frac{\gamma}{\gamma - 1}} \]
Substituting the known values:
\[ \frac{P}{27P} = \left(\frac{300}{T_2}\right)^{\frac{3/2}{3/2 - 1}} \]
This simplifies to:
\[ \frac{1}{27} = \left(\frac{300}{T_2}\right)^{3} \]

### Step 4: Rearranging the Equation
Now we can rearrange the equation to isolate \( T_2 \):
\[ \left(\frac{300}{T_2}\right)^{3} = 27 \]
Taking the cube root of both sides:
\[ \frac{300}{T_2} = 3 \]

### Step 5: Solve for \( T_2 \)
Now, we can solve for \( T_2 \):
\[ T_2 = \frac{300}{3} = 100 \, K \]

### Step 6: Final Result
Thus, the final temperature \( T_2 \) after adiabatic compression is:
\[ T_2 = 100 \, K \]