2.1 Ideal gas equation of state

We begin by discussing the properties of an ideal gas in the thermodynamic limit. The thermodynamic limit is defined as letting the volume approach infinity while maintaining a constant temperature (and chemical potential). More meaningfully for this discussion, it is the limit in which the size and shape of the container about the gas have no influence on its properties. There are many more molecules in the bulk of the gas than in contact with any surface, and the walls of the container are far apart compared to the size of the molecule. We shall see later that the influences of breaking such a limit are the hallmarks of nanoscience. An ideal gas obeys the ideal gas law

(2.1)

where p is the pressure, V the volume, n the number of moles, R the gas constant, and T the temperature. Equation (2.1) is also an equation of state. It tells us how the state of the gas changes as the conditions of the gas change. It is a mathematical expression of the experimental fact that, for a given amount of any fluid (n moles of gas or liquid, ideal or not), any two of the variables p, V, and T constitute a complete set of independent variables. In other words, all fluids regardless of their microscopic structure have an equation of state f(p, V, T) = 0. As we shall see shortly, T is a proxy variable for molecular energy. Therefore, fluids other than ideal ones, which possess intermolecular interactions that influence their energy, will have different equations of state. Deviations from the ideal gas law will be greatest for those species that interact most strongly.

The ideal gas law can be derived from statistical mechanics. However, it can also be derived empirically and independently by developing an equation that is consistent with experimental results. The following experimental observations have been made in the limit of low pressure (the dilute gas limit), that is, they hold for ideal gases:

Boyle’s law

(2.2)

Charles’s/Gay-Lussac’s law

(2.3)

(2.4)

which implies

(2.5)

Avogadro’s principle states that

(2.6)

Therefore, we can construct an equation, the ideal gas law Eq (2.1), which is consistent with all of these conditions. The constant that makes the ideal gas law consistent is R, the (universal) gas constant. R is determined by finding the value of the following expression in the limit of low pressure

(2.7)

Note that the Boltzmann constant is defined as the gas constant per molecule; thus, division by Avogadro’s constant yields its value,

(2.8)

and the ideal gas law can be written either on a per mole or per molecule basis

(2.9)

Once we have the ideal gas law, the laws of Boyle and Charles and Gay-Lussac are superfluous as they can all be derived from pV = nRT. In other words, it is a good rule to learn the most fundamental equation and then manipulate it to derive other useful results. As an example, introducing the mass density ρ defined by

(2.10)

where m is the molecular mass, M the molar mass and V the volume, we can rewrite the ideal gas law as

(2.11)

2.1.1 Example

At 500 °C and 93.2 kPa, the mass density of sulfur vapor is 3.710 kg m⁻³. What is the molecular formula of sulfur under these circumstances?

Use the ideal gas law and the definition of density. First, rearrange the definition of density ρ

to arrive at an expression for the molar mass M,

Then rearrange the ideal gas law to obtain an expression for V/n

Now substitute the expression for V/n into the expression for M

The atomic weight of S is 32 g mol⁻¹; 256/32 = 8, so therefore the molecular formula is S₈.

2.2 Molecular degrees of freedom

All gas-phase molecules in a macroscopic container, regardless of whether they are monatomic, diatomic or polyatomic, can translate in three directions. In other words, they have three translational degrees of freedom. In macroscopic containers at temperatures of interest to any form of usual chemistry, the translational degrees of freedom are completely classical. By that we mean that they are accurately described by the laws of classical mechanics. In principle, they can also possess electronic excitations, although electronic states are generally separated by large differences in energy. Usually, molecules are in their ground electronic state (the ground state is the lowest energy state). Only at very high temperatures are a significant number of molecules in electronic excited states (excited states are states that are at higher energy than the ground state), so we need not consider them further here. Similarly, we will only consider nuclear degrees of freedom when we deem them necessary, and will not discuss them in this general case. For a monatomic gas that is the end of the story as far as degrees of freedom are concerned.

Diatomic and polyatomic molecules possess bonds between atoms. Molecules are free to rotate in space and the bonds can vibrate. Each atom has three degrees of freedom before it enters the molecule. A molecule with N atoms must therefore have 3N degrees of freedom. Three of these degrees are translational, while the remaining (3N − 3) degrees correspond to internal degrees of freedom of rotation and vibration. A diatomic molecule can only vibrate in one way, in that the bond can stretch and contract. The two remaining degrees are rotational degrees of freedom. Think of a pencil. You can rotate it in the plane of the table or perpendicular to the plane of the table. Any other orientation of rotation can be constructed as a combination of in-plane and out-of-plane components. Rotation along the internuclear axis is of no consequence. The same partitioning of three translations and two rotations holds for any linear polyatomic molecule, which leaves (3N − 5) vibrational degrees of freedom. A nonlinear polyatomic molecule has three translational degrees as well as three rotational degrees of freedom (x, y, and z components to both translational and rotational motion). The remaining (3N − 6) degrees are vibrational degrees of freedom. As the molecular size increases, the number of available vibrational states increases rapidly. As a first approximation, we will consider that the degrees of freedom are independent of each other.

Rotations and vibrations are inherently quantum mechanical on a molecular level. We will explore this in detail in later chapters, but for now it will suffice to describe the broad details of the models we use to describe rotating and vibrating molecules.

The first-order approximation to a rotating diatomic molecule is a rigid rotor (Fig. 2.1). A rigid rotor is composed of a massless rod of fixed length corresponding to the equilibrium bond distance R_e. This rod connects two point masses m₁ and m₂. Quantum mechanically, the rigid rotor is only allowed to attain certain values of energy; that is, the rotational energy level structure is quantized. So, why is rotation quantized but translation is not? Translation occurs on the distance scale of the container in which the molecule resides. As long as this container is large compared to the molecule, translation acts classically and has a continuous distribution of energy levels. As we shall see later, we can model rotation as the motion of a particle on a sphere (or spheroid) of molecular dimensions. Rotation is confined to a distance of molecular dimensions and is, therefore, quantized. Confinement to small dimensions is inherent for bringing out quantized behavior.

The energy level spacing is larger for lighter masses and shorter bonds. Thus, H₂ has much larger energy spacings than I₂. The energy of a rigid rotor is

(2.12)

with a rotational constant B defined by

(2.13)

where J = 0, 1, 2, … is the rotational quantum number, h is Planck’s constant, ℏ = h/2π (pronounced h bar and called the reduced Planck constant), and c is the speed of light. (B is conventionally given in cm⁻¹ rather than the SI unit of m⁻¹.) By using a value of c = 3.00 × 10¹⁰ cm s^{− 1}, the unit cm is cancelled in Eq. (2.12) and the energy is given in J. Note that J only takes on discrete values as expected for a quantized variable, rather than a continuous distribution of values as would be expected for a classical variable. The reduced mass is defined by

(2.14)

The energy level spacing between two successive levels is given by

(2.15)

Since B = 0.03737 cm⁻¹ and 60.85 cm⁻¹ for I₂ and H₂, respectively, these spacings are on the order of 0.1–100 cm⁻¹ = 0.012–1.2 kJ mol⁻¹ = 0.12–12 meV per molecule. This compares to k_BT = 25 meV at room temperature. Thus, at sufficiently high temperatures many rotational levels are populated in heavy molecules, and the distribution of molecules that populate these states is approximated by a classical continuous distribution. Diatomic hydrides and lighter molecules have significantly fewer, discretely populated levels (we will discuss this distribution in detail later). Equation (2.15) shows that the energy level spacing increases with increasing J. However, because space is also quantized (or at least the direction into which the rotational angular momentum is quantized), each level contains (2J + 1) levels at the same energy (corresponding to the (2J + 1) directions into which the rotational angular momentum vector can point). We say that each level has a degeneracy of (2J + 1) and that levels at the same energy are degenerate. Since both energy level spacing and degeneracy are increasing functions of J, we can ask how the density of states increases – that is, how does the number of states per unit energy change? The density of states ρ(J) is defined (in the limit of large J) by

(2.16)

Equation (2.16) shows that the rotational density of state is constant.

2.2.1 Directed practice

Perform the conversions of energy units to show that 0.1–100 cm⁻¹ = 0.012–1.2 kJ mol⁻¹ = 0.12–12 meV.

The harmonic oscillator (Fig. 2.2) is the first approximation for the vibrational motion of a bond. The harmonic oscillator energy levels are quantized according to

(2.17)

where the fundamental wavenumber (again in cm⁻¹) depends on the force constant k and reduced mass μ according to

(2.18)

The vibrational quantum number v again takes on integer values v = 0, 1, 2…. Larger values of k and correlate with stiffer, stronger bonds. Again, H₂ with the highest fundamental wavenumber is anomalously high. I₂ exhibits a fundamental wavenumber of only 215 cm⁻¹. Bonds tend to be harmonic in the ground state, but at higher and higher levels of excitation the energy levels are spaced progressively closer together and eventually the bond breaks. In other words, the density of states of a real oscillator increases with increasing vibrational energy. These aspects of anharmonicity are better represented by the Morse oscillator energy level structure shown in Fig. 2.2(c). To a first approximation, we consider each vibrational degree of freedom as corresponding to its own independent oscillator.

Analogous to the rotational energy distribution, there is a high-temperature limit in which vibrational energy will be well approximated by a classical continuous distribution. This is most likely to be reached for polyatomic molecules containing heavy atoms. Since vibrational energy spacings are typically much larger than rotational energy level spacings, quantum mechanical corrections to classical expectations are much more likely when vibrations need to be considered.

2.3 Translational energy: Distribution and relation to pressure

The kinetic hypothesis states that the molecules that make up matter are constantly in motion, even when body of matter is at rest. Here, we consider the translational motion of an ideal gas, also known as a perfect gas. That molecules are translating means that they have both kinetic energy () and linear momentum (mv), both of which depend on the molecular mass m and its velocity v. Momentum-changing collisions of gas molecules with the container walls cause a force to be exerted on the walls. Pressure is a measure of this force per unit area. Thus, the mass and velocity of the molecules contribute to both the internal energy of the sample and its pressure. To simplify matters further we will assume that our perfect gas is not only colliding elastically (no internal energy changes including electronic excitation) and composed of noninteracting particles, but also that it is dilute (far from condensing) and composed of monatomic particles.

Consider a gas that is isolated from the rest of the universe (no exchange of energy or matter) such that the macroscopic properties are not changing over time. We state that this system is ‘at equilibrium.’ When averaged over time, its temperature (which we have yet to define) and pressure are constant. Both of these quantities are dependent on the velocity distribution of the gas molecules; therefore, the time average of the velocity distribution must also be constant. This does not mean that the velocity of each individual gas molecule is constant. On the contrary, the molecules are continually colliding with each other and exchanging momentum. However, because there is a huge number of molecules, say, on the order of Avogadro’s number, the individual fluctuations experienced by each molecule average out. There will be fluctuations in the velocity distribution, but the magnitude of these fluctuations tends to be small compared to the mean value averaged over the whole distribution. This in turn means that the fluctuations in p and T are small compared to their mean values. At equilibrium, the gas is also uniformly distributed throughout the container and the container is sufficiently large such that any property measured is independent of the shape of the container or the position in the container at which a measurement is performed. Thus, for instance, Pascal’s law – that the pressure is equal in all directions – holds at equilibrium.

The distribution of molecular velocities and positions is not only homogeneous (uniform in space) but also isotropic (uniform in all directions) at equilibrium. This means that there are no net flows in the gas, which requires that the average (indicated by angle brackets) of velocity vectors must vanish

(2.19)

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