The temperature at which Methane molecules have the same average Kinetic energy as that of oxygen molecules at 27°C is

Options

A. `327^(@)` C
B. `27^(@)` C
C. `927^(@)` C
D. `627^(@)` C

Answer

To find the temperature at which methane molecules have the same average kinetic energy as that of oxygen molecules at 27°C, we can use the concept of average kinetic energy of gas molecules.

### Step-by-Step Solution:

1. **Understanding Kinetic Energy**:
The average kinetic energy (KE) of a gas molecule is given by the formula:
\[ KE = \frac{3}{2} k T \]
where \( k \) is the Boltzmann constant and \( T \) is the absolute temperature in Kelvin.

2. **Convert the Given Temperature**:
The temperature of oxygen is given as 27°C. We need to convert this to Kelvin:
\[ T_{O_2} = 27 + 273.15 = 300.15 \, K \]

3. **Calculate the Kinetic Energy of Oxygen**:
Using the formula for kinetic energy, we find the kinetic energy of oxygen at this temperature:
\[ KE_{O_2} = \frac{3}{2} k (300.15) \]

4. **Setting Up the Equation for Methane**:
We want to find the temperature \( T_{CH_4} \) at which methane has the same kinetic energy as oxygen:
\[ KE_{CH_4} = \frac{3}{2} k T_{CH_4} \]
Since we want \( KE_{CH_4} = KE_{O_2} \), we can set the equations equal to each other:
\[ \frac{3}{2} k T_{CH_4} = \frac{3}{2} k (300.15) \]

5. **Solving for \( T_{CH_4} \)**:
We can cancel out \( \frac{3}{2} k \) from both sides:
\[ T_{CH_4} = 300.15 \, K \]

6. **Convert the Temperature Back to Celsius**:
To find the temperature in Celsius:
\[ T_{CH_4} = 300.15 - 273.15 = 27 \, °C \]

### Final Answer:
The temperature at which methane molecules have the same average kinetic energy as that of oxygen molecules at 27°C is **27°C**.