Partition functions monatomic gases

The standard entropies of monatomic gases are largely determined by the translational partition function, and since dris involves the logarithm of the molecular weight of the gas, it is not surprising that the entropy, which is related to tire translational partition function by the Sackur-Tetrode equation,... 

Using expression (38) of Chapter 5 for its partition function and U0 = 0, find a formula for the chemical potential of an ideal monatomic gas. Show that (9p/9P)r = Vm. 

For a system of non-interacting monatomic particles (an ideal gas) the microcanonical partition function is proportional to VN. Based on Q, VN, we can derive the state equation known as the ideal gas law ... 

The adsorbate is now visualized as a two-dimensional gas that can move freely tangential to the surface, where the molecules are "caught" In the force field of the surface. They have potential energy (caused by this attraction), vibration energy (normal to the surface) and translational energy. In (1.3.5.15) the following equation was derived for the sub-system partition function of a monatomic adsorbate ... 

Utilize the expression for the partition function derived in Chapter VI to develop an equation for the free energy of an ideal, monatomic gas referred to the value Fo in the lowest energy state. 

Using the partition function, show that for a monatomic gas, U = jNkT and that p = NkT/V. 

Information from Section 28-5 can be used to calculate the internal energy and heat capacity of a monatomic gas because the complete temperature dependence of Z is accounted for by partition functions for translational motion and the electronic ground state. Molecules exhibit 3 degrees of freedom per atom. Hence, there are no internal degrees of freedom for a monatomic gas (i.e.. He, Ne, Ar, Kr, Xe) because all 3 degrees of freedom are consumed by translational motion in three different coordinate directions. The internal energy is calculated from equation (28-59) ... 

Internal Partition Function for Monatomic Gases.—For the present purpose, for the internal energy of a monatomic gas only the nuclear spin and electronic states need be considered. On the assumption that these energies are additive, the partition function can again be factored ... 

The partition function of a monatomic gas is a product of three separate partition functions defined by the translational energy levels, the electronic energy levels, and the nuclear energy levels ... 

Further, we will presume at this point that the translational partition function, qtrans. is the major contributor to the thermodynamic properties of a monatomic gas. (We will justify this by using the kinetic theory of gases, which is covered in Chapter 19. The relative contributions of q iect and q will be considered in Chapter 18.) Therefore, for a monatomic ideal gas, we are assuming that... 

Once we establish the complete partition function of a molecular gaseous species, we will consider one additional application of the partition function the chemical change. In the last chapter, a few exercises asked for a determination of the A(something) of a physical process, like the expansion of a monatomic gas. However, in chemistry we are often concerned with the change in the chemical identity of a species—a chemical reaction. It may surprise you to learn that the partition functions of each chemical species in a balanced chemical reaction can be used to determine a characteristic property of that reaction its equilibrium constant. 

In chapter 17, we suggested that the overall partition function q for a monatomic gas is... 

First, we will state that even though the exact expression for the partition function Q is somewhat expanded from the partition function q for a monatomic gas, the basic relationships between Q and various thermodynamic functions are the same. That is. 

There is also a simple relationship between the pressure of a monatomic gas and its kinetic energy, which can be considered solely as energy of translation. (We are ignoring electronic and nuclear energies, as we did in our original discussion of partition functions of monatomic gases.) Because the classical expression for kinetic energy is... 

Curiously, a part of this expression is very similar to part of the translational partition function of a monatomic gas, and is proportional to the thermal de Broglie wavelength (see Chapter 17). 

Write the translational partition function for a monatomic dilute gas at 298.15 K and a volume of 0.0244 m as a constant times where M is the molar mass in kg moP Evaluate the partition function for He, Ne, Ar, Kr, and Xe. Explain the dependence on molar mass. 

Wehave already determined that the molecular partition function for a dilute monatomic gas is the product of a translational partition function and an electronic partition function. We obtained a formula for the translational partition function in Eq. (25.3-21) ... 

Since the logarithm of the partition function is a sum of terms, the internal energy of a dilute gas is a sum of conttibutions. For a monatomic substance... 

For a monatomic gas without electronic excitation Eq. (25.3-18) gives the formula for the molecular partition function ... 

As previously stated, the classical molecular partition function has units of kg s raised to some power, so a divisor with units must be included to make the argument of the logarithm dimensionless. If a divisor of lkgm s is used, values are obtained for the entropy and the Helmholtz energy that differ from the experimental values. However, when the classical canonical translational partition function is divided by h A and Stirling s approximation is used for ln(iV ), the same formulas are obtained as Chapter 26. For a dilute monatomic gas the corrected classical formula is... 

Q, the partition function, is specific to the system under consideration. The analysis in Chapter 1 determined Q to within a proportionality constant, C, for ideal monatomic gas particles Q = C(2KmkTy according to Equation 1.34. The distribution law can give the average number of particles that possess a given discrete energy, but for fhe gas kinetic analysis needed here, fhe distribution needs to be expressed in terms of ofher values such as velocity components, momentum, or speed. 

---