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The most basic physical principles in the vacuum industry :kiss: There are specific physical principles that serve as the foundation for the working mechanism of any vacuum product or vacuum process. However, what will be covered in this lecture are only the most basic physical principles that are frequently encountered in various fields of the vacuum industry. It mainly includes the properties of gases and vapors as well as the laws governing various dynamic processes within them (space processes), the laws of interaction between gases and solids (wall processes), and the laws governing gas flow. Due to space constraints, only a brief introduction can be provided for each aspect. I. Law of Ideal Gases It should be noted first that the laws and formulas introduced in this section and the following sections are derived for ideal gases in a state of equilibrium. However, various gases at normal temperature (compared to room temperature) and low pressure (relative to atmospheric pressure) can be regarded as ideal gases to a fairly good extent; therefore, we can confidently apply these laws and formulas to the vast majority of calculations in vacuum engineering. This includes various gases that are typically involved, as well as steam that is nearly saturated (such as water vapor) ； It also includes various gas-phase processes, even obvious non-equilibrium states (such as gas flow processes). The relationships between the state parameters of a gas, such as its pressure p (Pa), volume V (m3), temperature T (K), and mass m (kg), are governed by the following gas laws: 1. Boyle’s-Mariotte’s law: For a fixed mass of gas, if the temperature remains constant, the product of the gas’s pressure and volume is constant: pV = constant (1). 2. Gay-Lussac’s law: For a fixed mass of gas, if the pressure remains constant, the volume of the gas is proportional to its absolute temperature: V/T = constant (2). 3. Charles’s law: For a fixed mass of gas, if the volume remains constant, the pressure of the gas is proportional to its absolute temperature. p/T = constant (3) The above three formulas* are commonly referred to as the three laws of gases. The specific application method is often to deal with two gas states connected by a constant-process relationship, where the fourth parameter is determined given three known parameters. For example, for a gas with initial pressure and volume of P1 and V1, if its volume increases to V2 as a result of isothermal expansion, then according to Boyle’s law and Charles’s law, the pressure of the gas after expansion can be calculated as P2 = P1V1/V2. This is precisely the most basic pumping principle of various positive-displacement vacuum pumps. 4. Dalton’s law: The total pressure of a mixture of gases that do not react chemically with each other is equal to the sum of the partial pressures of each gas. P = P1 + P2 + …… + Pn (4) The partial pressure of a particular component in a mixed gas refers to the pressure that this gas would exert if it were present alone. Dalton’s law states the property of mutual independence and linear superposition of the pressures of individual component gases. 5. Avogadro’s law: Gases of any kind, when in equal volumes, have the same number of molecules at the same temperature and pressure ； In other words, at the same temperature and pressure, gases of different types with the same number of molecules occupy the same volume. The number of molecules in 1 mole of any gas is called Avogadro’s number, NA = 6.022×1023 mol-1. Under standard conditions (po = 1.01325×105 Pa, To = 0°C), the volume of 1 mol of any gas is called the standard molar volume, Vo = 2.24×10-2 m3/mol. Based on the aforementioned gas laws, the ideal gas law can be derived, which expresses the quantitative relationship among the state parameters of a gas: p, V, T, and m. This law is given by pV = m/M(RT) (5), where M represents the molar mass of the gas in kg/mol, and R is the universal gas constant, with a value of 8.31 J/(mol·K). Given any three of the four parameters p, V, T, and m, the other value can be determined using this formula. For example, the mass of a gas, m, is given by m = pVM/(RT). For a fixed mass of gas, when it passes through any thermodynamic process (which does not have to be a constant-pressure process) from one state (with parameters p1, V1, T1) to another state (with parameters p2, V2, T2), the equation of state allows us to derive the relationship: p1V1/T1 = p2V2/T2 (6). By manipulating equation (5), it is also possible to calculate the number of gas molecules per unit volume, namely the number density n (in m-3), as well as the gas density p (in kg/m3): n = mNA/MV = pNA/RT = p/kT (7); p = m/V = pM/RT (8). The constant k = R/NA = 1.38×1023 J/K is known as the Boltzmann constant. II. Foundations of the Kinetic Theory of Gas Molecules 1. In an ideal gas in equilibrium, the distribution of the thermal motion speeds of its molecules follows Maxwell’s speed distribution law. The probability that a gas molecule has a thermal velocity between v and v+dv is given by dN/N = F(v)dv = 4π(mo/2πkT)³/²·exp(−mv²/2kT)·v²dv. Here, F(v) is a continuous function of the velocity v (in m/s), and it is known as the velocity distribution function. mo = M/NA, which is the mass of a gas molecule in kilograms. Using the rate distribution function, three characteristic rates that reflect the intensity of molecular thermal motion can be calculated. The most probable velocity vm is the velocity with the highest probability among the various different thermal motion velocities of gas molecules, that is, the v value corresponding to the maximum value of F(v) ； The arithmetic mean of the thermal motion speeds of all gas molecules is called the arithmetic average speed v ； Adding up the squares of the speeds of all gas molecules, dividing by the total number of molecules, and then taking the square root gives the root mean square speed vs. 2. The basic formula for the pressure of an ideal gas establishes a quantitative relationship between the intensity of the microscopic thermal motion of gas molecules and the macroscopic gas pressure: P = 1/3(nmv²) = 1/3(pv²) (11). 3. The distance that a molecule in a gas travels between two successive collisions with other molecules is called the free path. Free paths vary in length significantly, but the statistical average of numerous free paths remains constant; this value is known as the mean free path, denoted by λ (m). The mean free path of gas molecules of a single type is (12–see below). For a mixed gas containing k components, in equation (13), σ represents the effective diameter of the gas molecules (in meters), while the subscripts l and j denote the parameters for the 1st and j-th gas components, respectively. The mean free paths λe and λi(m) for the motion of electrons and ions in a gas can also be defined. It should be emphasized that the mean free path of electrons or ions mentioned here refers to the distance traveled by an electron or ion as it moves through a gas between two successive collisions with gas molecules; collisions between electrons or ions themselves are not taken into account. Therefore, the formulas for calculating the average mean free path of electrons and ions involve only parameters related to gas molecules, and are independent of the spatial density of electrons or ions. (14)(15) 4. The value of a single free path length of a gas molecule is entirely random, but the length distribution of many free paths follows certain statistical laws. The probability that the free path of a gas molecule is greater than a given length χ is given by equation (16). Similarly, the probability that the free path of an electron or ion moving in a gas is greater than a given length χ can be determined using equations (17) and (18). By utilizing these distribution laws, along with the formulas for the mean free path given in equations (12)–(15), it is possible to calculate the loss rate of a particle beam moving in a directed manner as it passes through gaseous space. Alternatively, the required degree of vacuum in such gaseous space can be determined based on the specified loss rate. For example, in an ion beam vacuum system, a high-energy ion stream is emitted from the ion source toward a target located 25 cm away. If it is required that the loss rate due to collisions between the ion stream and the residual gas molecules in the vacuum chamber be less than 5%, what should be the pressure of the residual gas at a temperature of 27°C? According to the given conditions, when χ = 0.25m, it is required that Pi(λi > χ) be between 1% and 5%. Using equation (18), we find that exp(-0.25/λi) must be at least 0.95; therefore, λi must be at least 0.25/(-ln0.95), which is equivalent to λi ≥ 4.87m. Substituting this result into equation (15) gives kT/πσ2p≥4.87m ； Taking the effective molecular diameter of air as σ=3.72 × 10-10 m, the required residual gas pressure is p≤1.38 × 10-23 × 300/(π×3.722×10-20×4.87), that is, p≤1.95 × 10-3 Pa. 5. Regarding the issue of gas molecule collisions with the solid surface in contact with them (such as container walls), it can be discussed from two aspects: the direction of incidence and the number of incidents. If the angle between a solid angle dw and the normal to the area element ds is θ, then the number of gas molecules dNθ that strike ds per unit time from the direction of dw is proportional to cosθ; this is what is commonly referred to as the cosine law: (19) The number of gas molecules that collide with a unit area of a solid surface per unit time is called the incident rate of gas molecules on the surface, denoted as ν (m-2s-1), and its formula is given by: (20) Under the assumption of equilibrium, the direction distribution and quantity of gas molecules leaving the solid surface should be consistent with those of the incoming molecules; therefore, calculations can still be carried out using equations (19) and (20). Knudsen’s law of cosine reflection also states that, regardless of the direction in which the gas molecules strike, their reflection follows the cosine rule given by equation (19). 6. If two connected vacuum vessels have different temperatures, the parameters of the gas inside when it reaches a state of equilibrium will also differ. Under low-vacuum conditions, that is, in a viscous flow regime, the equilibrium condition for the two containers is that the pressures are equal. The relationships among the gas pressure, temperature, and molecular number density in the two containers are given by: p1 = p2 and n1/n2 = T2/T1 (21). Under high-vacuum conditions, that is, in a molecular flow regime, the condition for dynamic equilibrium of the gases in the two containers is that the incident flux γ at the junction point is equal; thus, the relationship is as follows: (22). This phenomenon of gas flow caused by differences in temperature, which results in a pressure gradient at equilibrium, is known as heat flux drift. It will introduce errors in vacuum measurement. For example, if the temperature in the heat field area of a vacuum resistance furnace is 1800 K, and the vacuum gauge connected via a thin tube operates at a temperature of 300 K, then if the gauge measures a pressure of 2×10-4 Pa, the actual gas pressure inside the furnace can be calculated using equation (22). III. Steam: Steam (also known as a condensable gas) is a term used in contrast to inert gases (or non-condensable gases). For any gas, there exists a critical temperature; gases above this critical temperature cannot be liquefied through isothermal compression, and such gases are known as permanent gases ； Gases below the critical temperature can be liquefied simply by increasing pressure; these are vapors. The process by which vapor molecules in space return to the liquid is called condensation. The condensation rate W of steam, that is, the mass of steam that condenses per unit area on a liquid surface per unit time, can be calculated using equation (20). In equation (23), α is the condensation coefficient, and pv is the partial pressure of steam. The reverse process of condensation, that is, the phenomenon in which liquid molecules move into space and turn into vapor, is called evaporation. The mass of liquid that evaporates per unit area per unit time is called the evaporation rate Gv. In conditions where both vapor and liquid coexist, evaporation and condensation occur simultaneously; if the evaporation rate is greater than the condensation rate, it manifests macroscopically as the evaporation of the liquid ； If the evaporation rate is less than the condensation rate, it manifests macroscopically as the condensation of vapor ； When the two are equal, it is in a saturated state; at this point, the pressure of the vapor in the space is referred to as the saturated vapor pressure ps at the corresponding equilibrium temperature. The saturated vapor pressure of a substance increases as the temperature rises. The relationship between the evaporation rate of a liquid and its saturated vapor pressure at the corresponding temperature is given by equation (24). This formula is commonly used to calculate the amount of metal that evaporates during evaporation coating processes. The ratio of the actual pressure pv of a steam to its saturated vapor pressure ps at the corresponding temperature is called the saturation degree of that steam at that time. As the most commonly used indicator parameter, the saturation of water vapor in the air is often defined as the relative humidity of the air. Relative humidity (%) = pvH2O/psH2O × 100% (25). For example, in engineering, standard environmental conditions are defined as a temperature of 20°C, a relative humidity of 65%, and an atmospheric pressure of 101325 Pa. Given that the saturated vapor pressure of water vapor at 20°C is 2333 Pa (17.5 torr), the partial pressure of water in the atmosphere under standard conditions can be calculated as 0.65 × 2333 = 1516 Pa (11.375 torr). The existence of a saturated vapor pressure is what fundamentally distinguishes steam from the ideal gas model, and it is also the reason why we need to discuss the properties of steam in a separate section. In vacuum engineering, before the steam reaches saturation, that is, the saturation degree