A cubically shaped vessel 20 cm on a side constains diatomic H2 gas at a temperature of 300 K .

step 1 To start out, this problem gives us the following situation. We have a hydrogen molecule with a

A cubically shaped vessel 20 cm on a side constains diatomic...

A cubically shaped vessel 20 cm on a side constains diatomic $\mathrm{H}_2$ gas at a temperature of 300 K . Each $\mathrm{H}_2$ molecule consists of two hydrogen atoms with mass of $1.66 \times 10^{-24} \mathrm{~g}$ each, separated by $\sim 10^{-8} \mathrm{~cm}$. Assume that the gas behaves like an ideal gas. Ignore the vibrational degree of freedom. (a) What is the average velocity of the molecules? (b) What is the average velocity of rotation of the molecules around an axis which is the perpendicular bisector of the line joining the two atoms (consider each atom as a point mass)? (c) Derive the values expected for the molar heat capacities $C_p$ and $C_v$ for such a gas.

A cubically shaped vessel 20 cm on a side constains diatomic $\mathrm{H}_2$ gas at a temperature of 300 K . Each $\mathrm{H}_2$ molecule consists of two hydrogen atoms with mass of $1.66 \times 10^{-24} \mathrm{~g}$ each, separated by $\sim 10^{-8} \mathrm{~cm}$. Assume that the gas behaves like an ideal gas. Ignore the vibrational degree of freedom.
(a) What is the average velocity of the molecules?
(b) What is the average velocity of rotation of the molecules around an axis which is the perpendicular bisector of the line joining the two atoms (consider each atom as a point mass)?
(c) Derive the values expected for the molar heat capacities $C_p$ and $C_v$ for such a gas.

Key Concepts

Molar Heat Capacity

Molar heat capacity at constant volume (C_v) and at constant pressure (C_p) are derived from the contributions of each degree of freedom to the energy of the gas, using the equipartition theorem. For a diatomic gas (with ignored vibrational modes), the total number and type of active degrees of freedom (translational plus rotational) determine the specific heat capacities, with the relation C_p = C_v + R, where R is the gas constant.

Rotational Motion and Moment of Inertia

For diatomic molecules treated as two point masses, calculating the rotational motion involves determining the moment of inertia about the relevant rotational axis. The moment of inertia, which depends on the mass distribution relative to the rotation axis, is used to compute the rotational energy. The average rotational velocity can be derived by relating the rotational kinetic energy (given by 1/2 I ?^2) to the temperature via the equipartition theorem.

Degrees of Freedom

In molecular systems, degrees of freedom refer to the independent ways in which a molecule can store energy. For diatomic molecules, there are translational, rotational, and vibrational modes. However, if vibrational modes are neglected, only translational and rotational degrees of freedom contribute. The number of these active modes directly influences the internal energy and, consequently, the molar heat capacities.

Kinetic Theory and Root Mean Square Velocity

Kinetic theory links the macroscopic properties of gases with the microscopic motion of individual particles. The root mean square (rms) velocity is derived using the relation v_rms = ?(3k_BT/m), where k_B is Boltzmann's constant, T is the absolute temperature, and m is the mass of a molecule. This provides insight into the average speed of molecules in the gas at a given temperature.

Ideal Gas Behavior

This concept models a gas as a collection of non-interacting point particles that obey the ideal gas law (PV = nRT). It is used to simplify the analysis by assuming that the molecules have negligible volume and do not exert intermolecular forces, which allows the derivation of many properties of gases such as pressure, temperature, and velocity distributions.

Equipartition Theorem

The equipartition theorem states that, for a system in thermal equilibrium, each quadratic degree of freedom contributes (1/2)k_BT to the energy per molecule. When applied to gases, this theorem is used to relate molecular motion (both translational and rotational) to the internal energy, which in turn affects properties like the heat capacity.

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