Question

The accessibility of rotational degrees of freedom. Diatomic ideal gases at \(T=300 \mathrm{~K}\) have rotational partition functions of approximately \(q=200\). At what temperature would \(q\) become small (say \(q<10\) ) so that quantum effects become important?

Short answer

At temperatures below 15 K, the rotational partition function q will be less than 10.

Step-by-step solution

Understand the Given Data

We're given that the rotational partition function, denoted as \(q\), is 200 at a temperature \(T=300 \, \mathrm{K}\). We need to determine the temperature at which \(q\) becomes less than 10.

Rotational Partition Function Formula

The rotational partition function for a diatomic molecule is given by: \[ q = \frac{T}{\theta_{rot}} \] where \(\theta_{rot}\) is the characteristic rotational temperature.

Calculate Characteristic Rotational Temperature

Using the given value \(q = 200\) at \(T=300 \, \mathrm{K}\), solve for \(\theta_{rot}\):\[ \theta_{rot} = \frac{T}{q} = \frac{300}{200} = 1.5 \, \mathrm{K} \]

Find the Temperature for \(q

To find the temperature at which \(q < 10\), we use the same formula and solve for \(T\):\[ q < 10 \Rightarrow \frac{T}{1.5} < 10 \Rightarrow T < 10 \times 1.5 = 15 \, \mathrm{K} \]

Key concepts

Rotational Degrees of Freedom

In the context of molecular thermodynamics, **rotational degrees of freedom** refer to the number of independent ways a molecule can rotate. For diatomic molecules, such as oxygen or nitrogen, there are typically two principal axes of rotation. These are orthogonal to the bond axis connecting the two atoms.

Each degree of freedom contributes to the energy stored in the system, thereby influencing the overall thermodynamic properties. At higher temperatures, more rotational energy levels become accessible, leading to higher values of the rotational partition function, denoted as \(q\).

Understanding rotational degrees of freedom helps us calculate important parameters like the rotational partition function, which are essential for predicting the behavior of gases.

Quantum Effects

As temperatures decrease, **quantum effects** become increasingly important in determining the behavior of rotational energy levels. At very low temperatures, the energy levels are not continuously accessible; instead, they become quantized. This means that only specific rotational states are energetically permissible.

In this exercise, quantum effects become significant when the rotational partition function \(q\) drops below 10. This indicates that fewer rotational states are accessible for the molecule, and it transitions from classical behavior (where many states are accessible) to quantum behavior (where fewer states are accessible).

Addressing these quantum effects is crucial for accurately modeling molecular behavior at low temperatures.

Characteristic Rotational Temperature

The **characteristic rotational temperature** \(\theta_{rot}\) is a fundamental property that influences the rotational behavior of a molecule. It is defined as the temperature at which the average rotational energy is comparable to the spacing between rotational energy levels.

In this exercise, it was determined that \(\theta_{rot} = 1.5 \text{ K}\) for a diatomic gas. The rotational partition function \(q\) is given by the formula \ q = \frac{T}{\theta_{rot}} \. This relationship tells us that \(q\) is directly proportional to the temperature \(T\).

To find the temperature at which \(q < 10\), we rearranged the formula and solved for \(T\). By doing so, we discovered that when \(T < 15 \text{ K}\), quantum effects dominate, and the rotational partition function becomes small. Understanding \(\theta_{rot}\) is key to predicting molecular behavior across different temperature ranges.