Which of the following ideal gases has higher value of average kinetic energy per mole at the same temperature - `N_2,CO_2,O_2` ?

Options

A. `N_2`
B. `CO_2`
C. `O_2`
D. All have equal value of KE

Answer

To determine which of the gases \( N_2 \), \( CO_2 \), or \( O_2 \) has a higher average kinetic energy per mole at the same temperature, we can follow these steps:

### Step 1: Understand the relationship between kinetic energy and temperature
The average kinetic energy per mole of an ideal gas is given by the formula:
\[ KE = \frac{3}{2} RT \]
where \( R \) is the universal gas constant and \( T \) is the temperature in Kelvin.

### Step 2: Recognize that kinetic energy is temperature-dependent
From the formula, we can see that the average kinetic energy per mole is directly proportional to the temperature \( T \). This means that if the temperature is the same for all gases, the average kinetic energy per mole will also be the same for all gases.

### Step 3: Analyze the given gases
We have three gases: \( N_2 \) (Nitrogen), \( CO_2 \) (Carbon Dioxide), and \( O_2 \) (Oxygen). Since the question states that they are all ideal gases at the same temperature, we can conclude that:

- The average kinetic energy per mole of \( N_2 \) at temperature \( T \) is:
\[ KE_{N_2} = \frac{3}{2} RT \]
- The average kinetic energy per mole of \( CO_2 \) at temperature \( T \) is:
\[ KE_{CO_2} = \frac{3}{2} RT \]
- The average kinetic energy per mole of \( O_2 \) at temperature \( T \) is:
\[ KE_{O_2} = \frac{3}{2} RT \]

### Step 4: Conclusion
Since all three gases are at the same temperature, their average kinetic energies per mole are equal:
\[ KE_{N_2} = KE_{CO_2} = KE_{O_2} \]
Thus, none of the gases has a higher average kinetic energy per mole than the others at the same temperature.

### Final Answer
All gases \( N_2 \), \( CO_2 \), and \( O_2 \) have the same average kinetic energy per mole at the same temperature.

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