For monoatomic gas, its initial temperature 127° C if adiabatically its volume changed to 8/27 of its initial value then what will be final temperature

Answer

To solve the problem step by step, we will use the principles of thermodynamics, specifically focusing on the adiabatic process for a monoatomic gas.

### Step 1: Convert the Initial Temperature to Kelvin
The initial temperature is given as 127°C. To convert this to Kelvin, we use the formula:
\[ T_i = 127 + 273 = 400 \, \text{K} \]

**Hint:** Remember that to convert Celsius to Kelvin, you need to add 273.

### Step 2: Identify the Initial and Final Volumes
Let the initial volume be \( V \). According to the problem, the final volume \( V_f \) is given as:
\[ V_f = \frac{8}{27} V \]

**Hint:** Identify the initial and final volumes clearly; this is crucial for using the adiabatic relations.

### Step 3: Use the Adiabatic Relation
For an adiabatic process, the relationship between temperature and volume can be expressed as:
\[ \frac{T_f}{T_i} = \left( \frac{V_i}{V_f} \right)^{\gamma - 1} \]
where \( \gamma \) is the adiabatic index.

### Step 4: Determine the Value of \( \gamma \)
For a monoatomic gas, the values of \( C_p \) and \( C_v \) are:
- \( C_p = \frac{5R}{2} \)
- \( C_v = \frac{3R}{2} \)

Thus, the value of \( \gamma \) is:
\[ \gamma = \frac{C_p}{C_v} = \frac{5/2}{3/2} = \frac{5}{3} \]

**Hint:** Remember that for monoatomic gases, you can easily find \( \gamma \) using the specific heat capacities.

### Step 5: Substitute Values into the Adiabatic Relation
Now we can substitute the known values into the adiabatic relation:
\[ \frac{T_f}{400} = \left( \frac{V}{\frac{8}{27}V} \right)^{\frac{5}{3} - 1} \]

This simplifies to:
\[ \frac{T_f}{400} = \left( \frac{27}{8} \right)^{\frac{2}{3}} \]

**Hint:** Simplifying the volume ratio is key to finding the final temperature.

### Step 6: Calculate the Volume Ratio
Calculating \( \left( \frac{27}{8} \right)^{\frac{2}{3}} \):
\[ \left( \frac{27}{8} \right)^{\frac{2}{3}} = \frac{27^{\frac{2}{3}}}{8^{\frac{2}{3}}} = \frac{9}{4} \]

### Step 7: Solve for Final Temperature
Now we can solve for \( T_f \):
\[ T_f = 400 \times \frac{9}{4} = 100 \times 9 = 900 \, \text{K} \]

**Hint:** Ensure you multiply correctly to find the final temperature.

### Final Answer
The final temperature of the gas after the adiabatic process is:
\[ T_f = 900 \, \text{K} \]