Question

The bond length of the \(\mathrm{N}_{2}\) molecule is \(0.1 \mathrm{nm},\) and its effective spring constant is \(2.3 \times 10^{3} \mathrm{~N} / \mathrm{m}\). (a) From the size of the energy jumps for rotation and vibration. deterinine whether either of these modes of energy storage should be active at \(300 \mathrm{~K}\). (b) According to the equipattivion theorem, the heat capacity of a diatomic molecule storing energy in rotations but not vibrations should be \(\frac{5}{2} R(3\) translational +2 rotational degrees of freedom). If it is also storing energy in vibrations. it should be \(\frac{7}{2} R\) (adding 2 vibrational degrees). Nitrogen's molar heat capacity is \(20.8 \mathrm{~J} / \mathrm{mol} \cdot \mathrm{K}\) at \(300 \mathrm{~K}\). Does this agree with your findings in part (a)?

Short answer

At $300K$, the rotational mode of the N2 molecule should be active but not vibrational. The molar heat capacity of Nitrogen molecule at $300K$ agrees with only rotational mode being active.

Step-by-step solution

Calculate Rotational Energy

We can calculate vibrational energy using the following formula: \(E_{rot} = J(J+1) \frac{kT}{2}\) where the constant J is the quantum number of the rotational state, for the lowest energy level, J equals 1. So, the energy is: \(E_{rot} = \frac{3kT}{2}\)

Determine Whether Rotational Mode is Active

To determine whether the rotational mode is active, compare \(E_{rot}\) with thermal energy \(kT = 300k\). Since rotational energy \(E_{rot}\) is less than thermal energy, rotational mode should be active at $300K$.

Calculate Vibrational Energy

The vibrational energy can be calculated using the following formula: \(E_{vib} = h\nu(n+\frac{1}{2})\) For the lowest energy level (n=0), the energy is: \(E_{vib} = \frac{1}{2}h\nu\). But \(\nu = \frac{1}{2\pi}\sqrt{\frac{k}{m}}\) where m is the reduced mass and k is the spring constant. Substituting the given values, we will get the value of the vibrational energy.

Determine Whether Vibrational Mode is Active

To determine whether the vibrational mode is active, compare \(E_{vib}\) with the thermal energy \(kT\). Vibrational energy is usually much greater than the thermal energy, so the vibrational mode is not likely to be active at $300K$.

Calculate Heat Capacity

Using the equipartition theorem we can calculate the heat capacity. If only rotations are happening, it should be \(\frac{5}{2}R\) but if vibrations are also active, it should be \(\frac{7}{2}R\).

Compare Calculated Heat Capacity and Given

By comparing the calculated heat capacity with the given (20.8 J/mol.K), we will strengthen our findings from part (a). The heat capacity should match better with the model that assumes only rotational degrees of freedom.

Key concepts

Rotational Energy

Rotational energy in diatomic molecules like \(_2\) arises from the motion of the molecules as they rotate around their center of mass. In simple terms, it's like how a spinning top gathers energy as it spins. This type of energy contributes to the molecule's internal energy.
To calculate rotational energy, we use the formula \(E_{rot} = J(J+1) \frac{kT}{2}\), where \J\ is the quantum number of rotational states. In practice, for the lowest energy level, \J\ is 1. By comparing this calculated rotational energy with a given thermal energy (\(kT\), where \Thermal energy: close to \(300k\)), we can determine if rotation contributes to the molecule's overall energy.
At 300 K, the rotational energy is typically less than the thermal energy, making it a significant factor in the molecule's dynamics.

Vibrational Energy

Vibrational energy, unlike rotational energy, is associated with the vibrations of atoms within the molecule. Imagine the atoms connected by a spring; the vibration is akin to stretching and compressing this spring. In the \(N_2\) molecule, this corresponds to the nitrogen atoms vibrating back and forth around a fixed point.
To compute vibrational energy, the formula \(E_{vib} = hu(n+\frac{1}{2})\) is applied. Its relevance can be tested against the thermal energy \(kT\). An intriguing aspect of vibrational energy is that it might be greater than the thermal energy, often rendering it inactive or negligible at moderate temperatures like 300 K. This contrasts with rotational energy, which frequently remains active under similar conditions.

Equipartition Theorem

The equipartition theorem is a crucial principle in molecular thermodynamics. It states that each degree of freedom contributes an equal amount of energy \(\frac{1}{2}kT\) to the total internal energy of the system. Degrees of freedom refer to independent ways in which a molecule's energy can be distributed or stored.
For diatomic molecules, such as \(N_2\), they have translational, rotational, and possibly vibrational degrees of freedom. This theorem helps predict the heat capacity of molecules based on which energy modes are active.

• If only rotations are considered, the heat capacity is refinable to \(\frac{5}{2}R\).
• If vibrations are added, the heat capacity increases to \(\frac{7}{2}R\).

This enhances our understanding of molecular behavior, especially concerning temperature and energy storage.

Heat Capacity

Heat capacity represents the amount of heat required to change a substance's temperature by a unit degree. For molecules, it greatly depends on energy modes, like translational, rotational, and vibrational movements present.
Quantifying this, we apply the equipartition theorem to predict how much heat is needed based on active modes.

• Diatomic molecules with only rotational and translational contributions yield a heat capacity approximated as \(\frac{5}{2}R\). Therefore, even a substance like nitrogen's specific heat capacity of 20.8 J/mol.K aligns closely with calculated predictive models.
• If vibrational modes are engaged, this number shifts to roughly \(\frac{7}{2}R\), implying a larger heat capacity.

These calculations are pivotal for understanding and designing experiments involving heat transfers in molecular substances.

Molecular Degrees of Freedom

Degrees of freedom in a molecule, such as \(N_2\), describe the independent directions or modes where the molecule can store energy.

• Translational energy, concerning movement along three dimensions (x, y, and z).
• Rotational energy, concerning rotation around two axes (for diatomic molecules).
• Vibrational energy in axes or planes where two atoms within a molecule can oscillate.

The total degrees of freedom determine how energy is partitioned amongst the available modes. For example, at room temperature, diatomic molecules primarily exhibit translational and rotational movements.
Understanding these degrees is essential as it helps explain molecular behavior across environments, supported by calculated heat capacities and the equipartition theorem.