Question

Evaluate the translational partition function for \(\mathrm{H}_{2}\) confined to a volume of \(100 . \mathrm{cm}^{3}\) at \(298 \mathrm{K}\). Perform the same calculation for \(\mathrm{N}_{2}\) under identical conditions. (Hint: Do you need to reevaluate the full expression for \(q_{T} ?\) )

Short answer

For \(\mathrm{H}_2\) confined in a volume of \(100\, \mathrm{cm}^3\) at \(298\, \mathrm{K}\), the translational partition function \(q_{T_{\mathrm{H}_2}}\) is approximately \(3.837 \times 10^{29}\). For \(\mathrm{N}_2\) under identical conditions, the translational partition function \(q_{T_{\mathrm{N}_2}}\) is approximately \(5.301 \times 10^{28}\).

Step-by-step solution

Convert volume from cm³ to m³

First, we need to convert the given volume from cm³ to m³. \[ 100 \, \text{cm}^3 = 100 \times (10^{-2})^3\, \text{m}^3 = 1.0 \times 10^{-4}\, \text{m}^3 \]

Calculate the mass of one molecule of \(\mathrm{H}_2\) and \(\mathrm{N}_2\)

Next, we need to find the mass of one molecule of \(\mathrm{H}_2\) and \(\mathrm{N}_2\). We will use their molar masses (MM) and Avogadro's number (NA) to calculate this: MM of \(\mathrm{H}_2\) = 2.016 g/mol MM of \(\mathrm{N}_2\) = 28.014 g/mol NA = \(6.022 \times 10^{23}\ \mathrm{mol}^{-1}\) First, convert the molar mass to kg/mol: MM of \(\mathrm{H}_2\) = 2.016 g/mol × (1 kg / 1000 g) = 0.002016 kg/mol MM of \(\mathrm{N}_2\) = 28.014 g/mol × (1 kg / 1000 g) = 0.028014 kg/mol Now, we can calculate the mass of one molecule: Mass of one \(\mathrm{H}_2\) molecule = \(\frac{0.002016\, \text{kg/mol}}{6.022\times10^{23}\, \mathrm{mol^{-1}}}=3.35 \times 10^{-27}\,\text{kg}\) Mass of one \(\mathrm{N}_2\) molecule = \(\frac{0.028014\, \text{kg/mol}}{6.022\times10^{23}\, \mathrm{mol^{-1}}}=4.65 \times 10^{-26}\,\text{kg}\)

Calculate the translational partition function for \(\mathrm{H}_{2}\) and \(\mathrm{N}_{2}\)

Now, we can use the formula for the translational partition function, which we derived in the analysis: \[ q_{T} = \frac{2 \pi m k_B T}{h^2}V \] Plug in the values for \(\mathrm{H}_2\): \[ q_{T_{\mathrm{H}_2}} = \frac{2 \pi (3.35 \times 10^{-27}\,\text{kg})(1.38 \times 10^{-23}\, \text{J/K})(298\, \mathrm{K})}{(6.63 \times 10^{-34}\, \text{Js})^2}(1.0 \times 10^{-4}\, \text{m}^3) \] \[ q_{T_{\mathrm{H}_2}} \approx 3.837 \times 10^{29} \] Now, do the same for \(\mathrm{N}_2\): \[ q_{T_{\mathrm{N}_2}} = \frac{2 \pi (4.65 \times 10^{-26}\, \text{kg})(1.38 \times 10^{-23}\, \text{J/K})(298\, \mathrm{K})}{(6.63 \times 10^{-34}\, \text{Js})^2}(1.0 \times 10^{-4}\, \text{m}^3) \] \[ q_{T_{\mathrm{N}_2}} \approx 5.301 \times 10^{28} \] We have successfully evaluated the translational partition function for both \(\mathrm{H}_{2}\) and \(\mathrm{N}_{2}\) under the given conditions: - \(q_{T_{\mathrm{H}_2}} \approx 3.837 \times 10^{29}\) - \(q_{T_{\mathrm{N}_2}} \approx 5.301 \times 10^{28}\)

Key concepts

Partition Function

In physical chemistry, the partition function is a cornerstone concept that is essential for understanding a system's thermodynamic properties. It is a sum over states that weights each possible state of the system by a factor that decreases exponentially with the energy of that state. Specifically, for a single molecule, the partition function can be expressed as

\[ q = \sum_{i} e^{ -\frac{E_i}{k_B T} } \]
where \( E_i \) is the energy of the \( i^{th} \) state, \( k_B \) is the Boltzmann constant, and \( T \) is the temperature. When we deal with an ideal gas, the partition function can be divided into different components corresponding to translational, rotational, vibrational, and electronic movements. The translational partition function, \( q_T \), gives information about the distribution of the molecules with respect to their movement through space. It reflects how molecules are spread out or confined within a particular volume at a given temperature. The translational partition function is crucial as it is directly related to pressure, volume, and temperature relationships in gases, which are central to many areas of chemistry and physics.

Molecular Mass

Molecular mass, sometimes referred to as molecular weight, is the mass of a single molecule of a substance and is typically expressed in atomic mass units (u) or grams per mole (g/mol). To move from the mass of a mole to the mass of a single molecule, Avogadro's number is used, which is the number of units in one mole of any substance. The molecular mass is a vital factor in the translational partition function because it directly affects the energy levels accessible to the molecule.

Knowing the molecular mass allows us to understand a substance's physical and chemical properties and how it will interact with other substances. It is also important in determining reaction stoichiometry and calculating concentrations. In the context of the translational partition function, the molecular mass is used to calculate the mass of one molecule that, in turn, is used to determine the energy distribution of the molecule's translational motion at a certain temperature.

Avogadro's Number

Avogadro's number, denoted as \( N_A \), is one of the fundamental constants in chemistry. It is defined as the number of atoms in exactly 12 grams of carbon-12, which is approximately \( 6.022 \times 10^{23} \) entities per mole. This constant allows chemists to bridge the gap between the atomic scale and the macroscopic scale by providing a link between the amount of substance in moles and the number of atoms or molecules.

In the context of calculating the translational partition function, Avogadro's number is instrumental in converting the molar mass of a gas into the mass of an individual molecule. This transformation is crucial since the energies involved in translational motion depend on the mass of the molecule. The more massive a molecule, the slower it moves for a given kinetic energy. Consequently, Avogadro's number not only simplifies counting atoms and molecules but also aids in contextualizing their behavior in large systems such as gases.

Physical Chemistry

Physical chemistry is the branch of chemistry that deals with the physical properties and transformations of materials, and how these relate to chemical structures and reactions. This field is where the principles of physics intersect with the molecular and atomic theories of chemistry. Topics in physical chemistry include thermodynamics, quantum chemistry, statistical mechanics, and kinetics.

At the heart of many physical chemistry problems, like the computation of the translational partition function, lies an understanding of the microscopic behavior of molecules and their interactions. It applies mathematics to describe and predict the properties of systems at the molecular level. Concepts like molecular mass and Avogadro's number are routinely utilized within this field to explain observations and experiments. By integrating these principles, physical chemists can derive relationships that govern the behavior of materials, advance technology, and deepen our comprehension of nature at the smallest scales.