A mono-atomic gas `X` and an diatomic gas `Y` both initially at the same temperature and pressure are compressed adiabatically from a volume `V` to `V//2`. Which gas will be at higher temperature?

Options

A. `X`
B. `Y`
C. same for both
D. can't say

Explanation

To determine which gas will be at a higher temperature after adiabatic compression, we can follow these steps:

### Step 1: Understand the Adiabatic Process
In an adiabatic process, there is no heat exchange with the surroundings. The relationship between temperature and volume for an ideal gas undergoing an adiabatic process can be expressed as:

\[ T_1 V^{\gamma - 1} = T_2 V^{\gamma - 1} \]

Where:
- \( T_1 \) is the initial temperature,
- \( T_2 \) is the final temperature,
- \( V \) is the volume,
- \( \gamma \) (gamma) is the heat capacity ratio \( \frac{C_p}{C_v} \).

### Step 2: Identify the Values of Gamma
For different types of gases, the value of \( \gamma \) varies:
- For a monoatomic gas (Gas X), \( \gamma = \frac{5}{3} \approx 1.67 \).
- For a diatomic gas (Gas Y), \( \gamma = \frac{7}{5} = 1.4 \).

### Step 3: Set Up the Equation
Given that both gases are compressed from volume \( V \) to \( \frac{V}{2} \), we can substitute these values into the equation:

\[ T_1 V^{\gamma - 1} = T_2 \left(\frac{V}{2}\right)^{\gamma - 1} \]

### Step 4: Rearranging the Equation
Rearranging the equation gives us:

\[ \frac{T_2}{T_1} = \left(\frac{V}{\frac{V}{2}}\right)^{\gamma - 1} = 2^{\gamma - 1} \]

### Step 5: Calculate Final Temperatures for Both Gases
Now we can calculate the final temperatures for both gases:

1. **For Monoatomic Gas (X)**:
\[ \frac{T_{2X}}{T_1} = 2^{\frac{5}{3} - 1} = 2^{\frac{2}{3}} \]
Thus,
\[ T_{2X} = T_1 \cdot 2^{\frac{2}{3}} \]

2. **For Diatomic Gas (Y)**:
\[ \frac{T_{2Y}}{T_1} = 2^{\frac{7}{5} - 1} = 2^{\frac{2}{5}} \]
Thus,
\[ T_{2Y} = T_1 \cdot 2^{\frac{2}{5}} \]

### Step 6: Compare Final Temperatures
To determine which gas has a higher final temperature, we compare \( 2^{\frac{2}{3}} \) and \( 2^{\frac{2}{5}} \):

Since \( \frac{2}{3} > \frac{2}{5} \), it follows that:

\[ T_{2X} > T_{2Y} \]

### Conclusion
The monoatomic gas \( X \) will be at a higher temperature than the diatomic gas \( Y \) after adiabatic compression.

---