The molar heat capacity , `C_(v)` of helium gas is `3//2R` and is independent of temperature. For hydrogen gas, `C_(v)` approaches `3//2R` at very low temperature, equal `5//2R` at moderate temperature and is higher than `5//2 R` at high temperatures. Give a reason for the temperature dependence of `C_(v)` in case of hydrogen, in not more than two or three sentences.

Explanation

To understand the temperature dependence of the molar heat capacity \( C_v \) for hydrogen gas, we need to analyze the contributions from different degrees of freedom at various temperature ranges.

1. **At Very Low Temperatures**:
- Hydrogen gas (H₂) behaves as a diatomic gas, but at very low temperatures, the rotational and vibrational modes are not excited. Thus, the only contributions to the heat capacity come from the translational degrees of freedom. For a diatomic gas, the translational contribution is \( \frac{3}{2}R \).

2. **At Moderate Temperatures**:
- As the temperature increases, the rotational modes of the hydrogen molecule become excited. Each rotational degree of freedom contributes \( \frac{1}{2}R \) to the heat capacity. Since hydrogen has two rotational degrees of freedom, the total contribution from rotation is \( R \). Therefore, the total heat capacity at moderate temperatures becomes:
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