Indicate the correct statement for equal volumes of `N_(2)(g)` and `CO_(2)(g)` at `25^(@)C` and 1 atm.

Options

A. The average translational K.E. per molecule is the same for `N_(2)` and `CO_(2)`
B. The rms speed is same for both `N_(2)` and `CO_(2)`
C. The density of `N_(2)` is less than that of `CO_(2)`
D. The total translational K.E. of both `N_(2)` and `CO_(2)` is the same

Explanation

To solve the question regarding equal volumes of \(N_2(g)\) and \(CO_2(g)\) at \(25^\circ C\) and 1 atm, we need to analyze the properties of these gases under the given conditions.

### Step-by-Step Solution:

1. **Understanding the Conditions**:
- We have equal volumes of \(N_2\) and \(CO_2\) at the same temperature (25°C) and pressure (1 atm). According to Avogadro's law, equal volumes of gases at the same temperature and pressure contain an equal number of moles.

2. **Number of Moles**:
- Since the volumes are equal, the number of moles of \(N_2\) and \(CO_2\) is the same. This means:
\[ n_{N_2} = n_{CO_2} \]

3. **Average Translational Kinetic Energy**:
- The average translational kinetic energy per molecule of a gas is given by the formula:
\[ KE = \frac{3}{2} k T \]
- Where \(k\) is Boltzmann's constant and \(T\) is the absolute temperature in Kelvin. Since both gases are at the same temperature, their average kinetic energies will be the same.

4. **RMS Speed Calculation**:
- The root mean square (RMS) speed \(v_{rms}\) of a gas is given by:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
- Where \(R\) is the universal gas constant, \(T\) is the temperature in Kelvin, and \(M\) is the molar mass of the gas. Since the molar masses of \(N_2\) (28 g/mol) and \(CO_2\) (44 g/mol) are different, their RMS speeds will differ.

5. **Density Comparison**:
- The density (\(\rho\)) of a gas can be calculated using the formula:
\[ \rho = \frac{PM}{RT} \]
- Since the pressure \(P\) and temperature \(T\) are the same for both gases, the density will depend on the molar mass \(M\). Therefore, \(N_2\) will have a lower density than \(CO_2\) because its molar mass is less.

6. **Conclusion**:
- The total translational kinetic energy of both gases is the same because it depends only on temperature. However, the RMS speeds and densities are different due to the difference in molar masses.

### Final Statement:
The correct statement is that the total translational kinetic energy of both \(N_2\) and \(CO_2\) is the same, while their densities and RMS speeds differ.