## Theoretical background

### 1. Introduction

The term vacuum refers to the condition of an enclosed space that is devoid of all gases or other material content. It is not experimentally feasible to achieve a “perfect” vacuum, although one can approach this condition extremely closely. It is possible routinely to obtain a vacuum of 10-6 Torr and with more sophisticated techniques 10-10 Torr (1.3 x 10-13 bar or 1.3 x 10-8 Pa); it is even possible by special techniques to obtain a vacuum of 10-15 Torr, or about 30 molecules per cubic centimeter. One Torr, the conventional unit of pressure in vacuum work, is the pressure equivalent of a manometer reading of 1 mm of liquid mercury; 1 Torr = 1/760 atm = 1.333 x 10-3 bar = 133.3 Pa.

For Dewar flasks, metal evaporation apparatus, and most research apparatus, a vacuum of 10-5 to 10-6 Torr is sufficient; this is in the "high vacuum" range, while 10-10 Torr would be termed "ultrahigh vacuum". However, for many routine purposes a "utility vacuum" or "forepump vacuum" of about 10-3 Torr will suffice, and for vacuum distillations only a "partial vacuum" of the order of 1 to 50 Torr is needed

Many famous scientists performed investigations under vacuum conditions. You can find interesting information about the History of Vacuum science & information about terminology and technology . Most modern day experimental research in physical chemistry is performed with the use of some sort of vacuum system. Organic and inorganic chemists are also finding it is essential to conduct synthetic and kinetic work under controlled or reduced pressures. Vacuum systems vary widely in their size and complexity depending on the requirements of pumping speed and attainable vacuum. This experiment is designed to illustrate the purpose and use of the basic components found on typical vacuum apparatus.

### 2. Gas flow in tubes

In order to achieve a low pressure in a vacuum line, some air must be removed by pumping; as it is removed it must flow from one end of the tube to the other. The rate of flow of a gas, called the throughput Q, is defined as

(1)
 Q = P dV dt

where P is the pressure at which it is measured, and dV/dt is the volume flow rate. Notice that throughput does not have the same units as ordinary gas flow rate (unit volume/unit time). The units of throughput are [pressure]·[volume]/[time] or [energy]/[time], that is, L·atm·min-1 (sometimes Torr·L/sec or in SI units, Pa·m3·s-1, or J·s-1, or watts). The throughput depends on the resistance to flow and the pressure drop between the entrance and exit to a tube or channel:

(2)
 Q = P2-P1 = C(P2-P1) Z

where P1 is the downstream pressure (measured at the exit). P2 is the upstream pressure (measured at the entrance), Z is the resistance and C is the conductance. The conductance is the throughput per unit pressure difference between the tube entrance and exit. The units of conductance are the same that of volume rate or pumping speed, so conductance can be expressed in L/min, L/sec, m3/hour, etc. The quantities in Equations (1) and (2) are quite analogous to Ohm's Law, which relates the flow of current, I, through a resistance R under the influence of a potential difference, E:

(3)
 I = E R

Equation (3) is the Ohm's Law of gas flow through a tube: it relates the gas flow Q (throughput) through a tube of resistance Z under the influence of a pressure difference P2 – P1. Just as the resistance of electrical resistors in series is given by:

(4)
 RT = R1 + R2 + R3 + R4 + ...

and in parallel by:

(5)
 1 = 1 + 1 + 1 + ... RT R1 R2 R3

so the resistance to gas flow is given by

(6)
 ZT = Z1 + Z2 + Z3 + Z4 + ...

where ZT is the total resistance and Zi are the series resistances of the various components in a vacuum line, that is, the traps, baffles, stopcocks, and tubes of different diameters. It is somewhat more common in discussing gas flow to speak of conductance rather than resistance, so Equation (6) is frequently written:

(7)
 1 = 1 + 1 + 1 + ... CT C1 C2 C3
 Figure 1: Series and paralel resistance.

### 2.1. Viscous Flow Versus Molecular Flow

The nature of gas flow through a tube is quite different at low pressures than at high pressures. In addition, the flow characteristics depend on the flow rate and the geometry of the tube, pipe, or channel through which the gas flows. Three kinds of flow are recognized: turbulent. viscous (laminar), and molecular. The Reynolds number R is useful in expressing the boundary between turbulent and viscous flow; similarly, the Knudsen number Kn helps define the boundary between viscous and molecular flow. The Reynolds number, a dimensionless number, is defined as:

(8)
 R = a·ρ·U η

where a is the tube radius, ρ is the gas density, η is the viscosity, and U is the flow velocity across a plane in the tube, defined as:

(9)
 U = Q π·a2·P

The Knudsen number Kn is purely empirical and is defined as the ratio of the mean free path L to a "characteristic dimension" of the system, say, the radius of the tube:

(10)
 Kn = L a

The Knudsen number is also dimensionless. The mean free path is given by:

(11)
 L = 1 21/2·π·d2·N

where d is the molecular diameter and N is the number of molecules per unit volume. Since N is related to the pressure and temperature it is convenient to express the mean free path for air at 25°C as:

(12)
 L = 0.005 P

where L is the mean free path in centimeters and P is the pressure in Torr. This is useful for getting a rough value for the Knudsen number for a tube or bulb in a vacuum line, the dimensions of which are usually measured in centimeters. The rough ranges of flow are summarized in Table 1:

 Types of gas flow Flow Type Pressure Reynolds No. Knudsen No. Turbulent High >2200 - Viscous Medium <1200 <0.01 Molecular Low - >1.00
Table 1.

As an example, let us calculate the maximum pressure (torr) at which molecular flow is observed in a long glass tube 25 mm in diameter:
L=Knmaximum·a=1.0 · 1.25cm = 1.25cm
P=0.005/L=0.005/1.25 = 4·10-3 Torr

### 2.2. Viscous Flow

Above about 10-3 Torr, gas properties depend upon collisions between molecules, which occur much more frequently than between molecules and their container. At pressures below 10-3 Torr, viscosity is not a property of a gas, since collisions between molecules are infrequent. In the region of viscous flow, the Poiseuille equation gives the throughput through a straight tube of circular cross section:

(13)
 Q = π·d4 Pavg·(P2-P1) 128·η·ℓ

where d and ℓ are the tube diameter and length, η is the gas viscosity, and Pavg is the average of P2 and P1.If we combine equations, we obtain an equation for the viscous flow conductance in a tube of circular cross section:

(14)
 C = π·d4 Pavg 128ηℓ

Note that the most widely tabulated unit of viscosity is the CGS unit, the poise: 1 poise = 1 g·cm-1·sec-1.

 Figure 2: 1 poise is the movement of 1 gram, 1 cm in 1 sec.

The SI unit is pascal·second: 1 Pa·s = 1 kg·m-1·s-1. Thus 1 poise = 0.1 Pa·s. The viscosity of air at 25°C is 1.845·10-4 poise = 1.845·10-5 Pa·s. If d and l are given in centimeters and Pave in torr, then conductance of tube, C in L/sec for air at 25 °C is:

(15)
 C[L/sec] = 182 d[cm]4 Pavg[torr] ℓ[cm]

In Figure 3, because the mean free path L is small compared to the radius of the tube a, collisions are more frequent between molecules than they are between molecules and the walls of the container. Consequently, the properties of the gas are quite constant over several mean free paths and the gas acts like a continuous viscous fluid. The Knudsen number is defined as L/a and when L/a<0.01, the gas flow is considered viscous. When the mean free path is large compared to the diameter of the tube (high Knudsen number or L/a>1.0), the gas molecules collide with the walls of the container more frequently than with each other.

 Figure 3: Viscous and molecular flow. (i) Viscous flow (L/a<0.01). The molecules collide much more frequently with each other than with the wall of the containing tube. (ii)) Molecular flow (L/a>1.0). The molecules collide much more frequently with the walls of the containing tube than with each other.

Viscosity is then undefined, since there can be no shearing forces between layers of molecules nor is any momentum transferred between molecular layers. These are the conditions for molecular flow, shown in figure (2 ii). To compare these two figures. visualize the mean free path in comparison to the dimensions of the tube and to the average distance between molecules.

### 2.3. Pumping Speed in Fixed Pressures

The pumping speed S at any point in the vacuum system is defined by the ratio of the throughput Q to the pressure at that point:

(16)
 S = Q P

The units of S are [unit volume/unit time], the same as the units of conductance. To design a vacuum line properly, it is useful to know how the pumping speed SL in the line differs from the pumping speed of the pump SP. The effect of line resistance in reducing the pumping speed is given by:

(17)
 1 = 1 + 1 SL SP C

Notice that if the conductance equals the speed of the pump, the pumping speed in the line is just one half that of the pump since:

(18)
 1 = 1 + 1 = 2 ⇒ SL = SP SL SP C SP 2

Consequently, it is important to make the conductance of the line as large as possible in comparison to the speed of the pump. Only when C is infinite (C→∞ ) does SL = SP. For a real vacuum line, the resistance to pumping is the sum of the resistances of all the components (traps, baffles, stopcocks, etc.) that make up the line. This relationship is usually written in terms of conductances (C = 1/Z):

(17)
 1 = 1 + ∑ 1 SL SP Ci

where the Ci are the conductances of the components. Thus, it is important to minimize the number of components and not to clutter up the vacuum line with unnecessary or unused traps and stopcocks. The viscous conductance depends on the fourth power of the tube radius. Consequently, tubing between the fore pump and the diffusion pump should be of as large a diameter as possible. Reducing the radius by one half reduces the conductance by one sixteenth. The graphical dependence of the pumping speed SL in the line on the conductance C of the line for various pumping speeds Sp is shown in Figure 4.

 Figure 4: A schematic illustration of pumping speed in line (SL) versus conductance (C) at selected pumping speeds (SP). The pumping speed falls off as the conductance decreases.

### 2.4. Pumping Speed in Fixed Volume

Up to this point, pumping speed has been discussed for the case of fixed pressures. For the case of evacuating a fixed volume V, the throughput Q becomes —Vdp/dt (The negative sign is required because the pressure p in volume V is decreasing). Therefore, the pressure will obey the differential equation:

(18)
 S = Q = — V · d P P P d t

A general solution of this equation is complicated, since S varies with pressure. For the simple case, where S is considered constant over a range of pressures, one finds:

(19)
 P = P0exp(— S )t V

where P0 is the pressure at t=0. The length of time for the pressure in volume V to fall by one decade (power of 10) is then: t0.1=2.303(V/S)