Four cylinders contain equal number of moles of argon, hydrogen, nitrogen and carbon dioxide at the same temperature. The energy is minimum in

Options

A. argon
B. hydrogen
C. nitrogen
D. carbon dioxide

Answer

To determine which gas has the minimum internal energy among argon, hydrogen, nitrogen, and carbon dioxide, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Internal Energy Formula**:
The internal energy \( U \) of an ideal gas can be expressed as:
\[ U = \frac{F}{2} nRT \]
where \( F \) is the degrees of freedom, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature.

2. **Identify the Number of Moles and Temperature**:
The problem states that all four cylinders contain an equal number of moles and are at the same temperature. Thus, \( n \) and \( T \) are constant for all gases.

3. **Determine the Degrees of Freedom for Each Gas**:
- **Argon (Ar)**: Argon is a monoatomic gas, so it has 3 degrees of freedom (all translational).
- **Hydrogen (H₂)**: Hydrogen is a diatomic gas, so it has 5 degrees of freedom (3 translational + 2 rotational).
- **Nitrogen (N₂)**: Nitrogen is also a diatomic gas, so it has 5 degrees of freedom (3 translational + 2 rotational).
- **Carbon Dioxide (CO₂)**: Carbon dioxide is a linear triatomic molecule, so it has 6 degrees of freedom (3 translational + 2 rotational + 1 vibrational).

4. **Calculate the Internal Energy for Each Gas**:
Since \( n \) and \( T \) are the same for all gases, the internal energy will depend solely on the degrees of freedom:
- For Argon:
\[ U_{Ar} = \frac{3}{2} nRT \]
- For Hydrogen:
\[ U_{H_2} = \frac{5}{2} nRT \]
- For Nitrogen:
\[ U_{N_2} = \frac{5}{2} nRT \]
- For Carbon Dioxide:
\[ U_{CO_2} = \frac{6}{2} nRT \]

5. **Compare the Internal Energies**:
- Argon has the lowest internal energy because it has the least degrees of freedom (3).
- Hydrogen and Nitrogen have the same internal energy (5).
- Carbon Dioxide has the highest internal energy (6).

6. **Conclusion**:
Therefore, the gas with the minimum internal energy is **Argon**.