A diatomic gas initially at `18^(@)` is compressed adiabatically to one- eighth of its original volume. The temperature after compression will b

Options

A. `10^(@)`
B. `887^(@)`
C. 668k
D. `144^(@)`

Answer

To solve the problem of finding the final temperature of a diatomic gas after it is compressed adiabatically to one-eighth of its original volume, we can follow these steps:

### Step 1: Understand the Initial Conditions
The initial temperature of the gas is given as \( T_i = 18^\circ C \). To convert this to Kelvin, we use the formula:
\[ T_i(K) = T_i(°C) + 273 = 18 + 273 = 291 \, K \]

### Step 2: Identify the Nature of the Process
The gas is compressed adiabatically, meaning there is no heat exchange with the surroundings. For an adiabatic process, we can use the formula:
\[ T_i \cdot V^{\gamma - 1} = T_f \]
where \( T_f \) is the final temperature, \( V \) is the initial volume, and \( \gamma \) is the heat capacity ratio.

### Step 3: Determine the Value of \( \gamma \)
For a diatomic gas, the specific heat capacities are:
- \( C_p = \frac{7R}{2} \)
- \( C_v = \frac{5R}{2} \)

Thus, the value of \( \gamma \) is:
\[ \gamma = \frac{C_p}{C_v} = \frac{7/2}{5/2} = \frac{7}{5} \]

### Step 4: Set Up the Equation for Final Temperature
Since the gas is compressed to one-eighth of its original volume, we have:
\[ V_f = \frac{V_i}{8} \]
Substituting this into the adiabatic condition gives us:
\[ T_i \cdot \left(\frac{V_i}{V_f}\right)^{\gamma - 1} = T_f \]
This can be rewritten as:
\[ T_f = T_i \cdot \left(\frac{V_i}{\frac{V_i}{8}}\right)^{\gamma - 1} = T_i \cdot (8)^{\gamma - 1} \]

### Step 5: Calculate \( \gamma - 1 \)
Calculating \( \gamma - 1 \):
\[ \gamma - 1 = \frac{7}{5} - 1 = \frac{2}{5} \]

### Step 6: Substitute Values into the Final Temperature Equation
Now we can substitute the values into the equation for \( T_f \):
\[ T_f = 291 \cdot (8)^{\frac{2}{5}} \]

### Step 7: Calculate \( (8)^{\frac{2}{5}} \)
Calculating \( (8)^{\frac{2}{5}} \):
\[ (8)^{\frac{2}{5}} = 2^{3 \cdot \frac{2}{5}} = 2^{\frac{6}{5}} \approx 2.297 \]

### Step 8: Final Calculation for \( T_f \)
Now, substituting back to find \( T_f \):
\[ T_f \approx 291 \cdot 2.297 \approx 668.5 \, K \]

### Conclusion
The final temperature after compression is approximately:
\[ T_f \approx 668.5 \, K \]