# Gas Phase

Matter can exist in three different physical forms, called phases or states: gas, liquid, and solid. The gaseous phase may be the simplest to understand because all gases display similar behavior and follow similar laws regardless of their particular chemical identities. Like liquids, gases are classified as fluids because they can flow and take on the shapes of their containers. However, the atoms or molecules in a gaseous sample move rapidly and are far apart from each other. In addition, only very weak intermolecular forces exist between gas particles; this results in certain characteristic physical properties, such as the ability to expand to fill any volume. Gases are also easily, although not infinitely, compressible, which distinguishes them from liquids.

Temperature is measured with a thermometer. The SI unit of temperature is the kelvin (K), measured on the Kelvin scale. However, temperature is more commonly measured on the Celsius scale (°C). The two scales are closely related. At standard pressure, water freezes at 0°C and boils at 100°C. The Kelvin scale increases by the same increments as the Celsius scale. The Kelvin scale starts at 0 K, which is absolute zero. This temperature is defined as the theoretical temperature at which all particle motion in matter stops. Absolute zero is –273.15°C. You can convert between the scales by adding or subtracting this value, which is often rounded to the nearest whole number.

K = 0C + 273

Medical devices that measure blood pressure are termed sphygmomanometers, and the most clinically relevant unit of measurement for them is mmHg. In fact many medical devices utilize the same conceptual design of a barometer, to continuously monitor blood pressure.

In order to explain why the mercury rises in a barometer, we must summarize the forces at play here. Atmospheric pressure creates a downward force on the pool of mercury at the base of the barometer while the mercury in the column exerts an opposing force (its weight) based on its density. The weight of the mercury creates a vacuum in the top of the tube. When the external air exerts a higher force than the weight of the mercury in the column, the column rises. When the external air exerts a lower force than the weight of the mercury, the column falls. Thus, a reading can be obtained by measuring the height of the mercury column (in mm), which will be directly proportional to the atmospheric pressure being applied.

When we examine the behavior of gases under varying conditions of temperature and pressure, we assume that the gases are ideal. An ideal gas represents a hypothetical gas with molecules that have no intermolecular forces and occupy no volume. Although real gases deviate from this ideal behavior at high pressures (low volumes) and low temperatures, many compressed real gases demonstrate behavior that is close to ideal.4  The ideal gas law shows the relationship among four variables that define a sample of gas:

## PV = nRT

where P is the pressure, V is the volume, n is the number of moles, and T is the temperature. R represents the ideal gas constant, which has a value of 8.314 J/Kmol-1. The ideal gas law is used to determine the missing term when given all of the others. It can also be used to calculate the change in a term while holding two of the others constant.

Robert Boyle conducted a series of experimental studies in 1660 that led to his formulation of a law that now bears his name: Boyle’s law. His work showed that, for a given gaseous sample held at constant temperature (isothermal conditions), the volume of the gas is inversely proportional to its pressure:

PV = k or P1V1 = P2V2

where k is a constant, and the subscripts 1 and 2 represent two different sets of pressure and volume conditions.4 Careful examination of Boyle’s law shows that it is, indeed, simply the special case of the ideal gas law in which n and T are constant.

Joseph Louis Gay-Lussac published findings based, in part, on earlier unpublished work by Jacques Charles; hence, the law of Charles and Gay-Lussac is more commonly known simply as Charles’s law.4 The law states that, at constant pressure, the volume of a gas is proportional to its absolute temperature, expressed in kelvin. Expressed mathematically, Charles’s law is:

V / T = K

Or V1T1 =V2T2

where, again, k is a proportionality constant and the subscripts 1 and 2 represent two different sets of temperature and volume conditions. Careful examination of Charles’s law shows that it is another special case of the ideal gas law in which n and P are constant.

Gay-Lussac’s law is complementary to Charles’s Law. It utilizes the same derivation from the ideal gas law, but it relates pressure to temperature instead.4 Expressed mathematically, Gay-Lussac’s law is:

PT = k

Or P1P1 = P2T2

where, again, k is a proportionality constant, and the subscripts 1 and 2 represent two different sets of temperature and pressure conditions. Careful examination of Gay-Lussac’s law shows that it is another special case of the ideal gas law in which n and V are constant.

The combined gas law was a combination of many of the preceding laws. This law relates pressure and volume (Boyle’s law) in the numerator, and relates the variations in temperature to both volume (Charles’s law) and pressure (Gay-Lussac’s law) simultaneously. When using this equation, take care to place all of the variables in the right place.4 The combined gas law is:

P1V1/T1 = P2V2/T2

Avogadro’s principle states that all gases at a constant temperature and pressure occupy volumes that are directly proportional to the number of moles of gas present. Equal amounts of all gases at the same temperature and pressure will occupy equal volumes.

N / V = k

Or n1/V1 = n2/V2

where k is a constant, n1 and n2 are the number of moles of gas 1 and gas 2, respectively, and V1 and V2 are the volumes of the gases, respectively. This can be summarized as the number of moles of gas increases, the volume increases in direct proportion.

The specific heat of a substance is the amount of heat required to raise the temperature of one mass unit by one degree Celsius. The following equation shows the relationship among the heat added to a substance Q, the specific heat c, the mass m, and the temperature T.

## Q= mcΔT

The equation holds true both for substances that absorb energy from their surroundings and those that transfer energy to their surroundings. When energy is absorbed, Q and ΔT are positive. When energy is transferred out of a substance, Q and ΔT. The term heat capacity is often used to describe a sample as a whole rather than as a unit mass. The heat capacity of a sample is:

Heat capacity = M x Q

The ideal-gas equation describes how gases behave but not why they behave as they do. A model that helps us to see what happens to gas particles when conditions such as pressure or temperature change is the kinetic-molecular theory of gases.5 The kinetic-molecular theory (the theory of moving molecules) is summarized by the following statements:

1. Energy can be transferred between molecules during collisions but, as long as temperature remains constant, the averagekinetic energy of the molecules does not change with time.
2. The combined volume of all the molecules of the gas is negligible relative to the total volume in which the gas is contained.
3. The average kinetic energy of the molecules is proportional to the absolute temperature. At any given temperature the molecules of all gases have the same average kinetic energy.
4. Gases consist of large numbers of molecules that are in continuous, random motion.
5. Attractive and repulsive forces between gas molecules are negligible.

The kinetic-molecular theory explains both pressure and temperature at the molecular level. The pressure of a gas is caused by collisions of the molecules with the walls of the container. The magnitude of the pressure is determined by how often and how forcefully the molecules strike the walls.5 The absolute temperature of a gas is a measure of the average kinetic energy of its molecules. If two gases are at the same temperature, their molecules have the same average kinetic energy (statement 5 of the kinetic-molecular theory). If the absolute temperature of a gas is doubled, the average kinetic energy of its molecules doubles. Thus, molecular motion increases with increasing temperature. According to the kinetic molecular theory of gases, the average kinetic energy of a gas particle is proportional to the absolute temperature of the gas:

KE = 1/2 mv2

KE = 3/2 KBT

where kB is the Boltzmann constant (1.38 x 10-23 JK-1) which serves as a bridge between the macroscopic and microscopic behaviors of gases. This means that is a bridge between the behavior of the gas as a whole and the individual gas molecules. This equation shows that the speed of a gas particle is related to its absolute temperature. However, because of the large number of rapidly and randomly moving gas particles, which may travel only nanometers before colliding with another particle or the container wall, the speed of an individual gas molecule is nearly impossible to define. Therefore, the speeds of gases are defined in terms of their average molecular speed. One way to define an average speed is to determine the average kinetic energy per particle and then calculate the speed to which this corresponds. The resultant quantity, known as the root-mean-square speed (urms), is given by the following equation:

urms = (3RT/M)

where R is the ideal gas constant, T is the temperature, and M is the molar mass.

The behavior of real gases usually agrees with the predictions of the ideal gas equation to within 5% at normal temperatures and pressures. At low temperatures or high pressures, real gases deviate significantly from ideal gas behavior. Dutch physicist Johannes van der Waals realized that two of the assumptions of the kinetic molecular theory were questionable. The kinetic theory assumes that gas particles occupy a negligible fraction of the total volume of the gas. It also assumes that the force of attraction between gas molecules is zero. The first assumption works at pressures close to 1 atm. But something happens to the validity of this assumption as the gas is compressed. Imagine for the moment that the atoms or molecules in a gas were all clustered in one corner of a cylinder. At normal pressures, the volume occupied by these particles is a negligibly small fraction of the total volume of the gas. But at high pressures, this is no longer true. As a result, real gases are not as compressible at high pressures as an ideal gas. The volume of a real gas is therefore larger than expected from the ideal gas equation at high pressures. Van der Waals proposed that we correct for the fact that the volume of a real gas is too large at high pressures by subtracting a term from the volume of the real gas before we substitute it into the ideal gas equation. He therefore introduced the constant (b) into the ideal gas equation that was equal to the volume actually occupied by a mole of gas particles. Because the volume of the gas particles depends on the number of moles of gas in the container, the term that is subtracted from the real volume of the gas is equal to the number of moles of gas times b.

## P(V – nb) = nRT

When the pressure is relatively small, and the volume is reasonably large, the nb term is too small to make any difference in the calculation. But at high pressures, when the volume of the gas is small, the “nb” term corrects for the fact that the volume of a real gas is larger than expected from the ideal gas equation. The assumption that there is no force of attraction between gas particles cannot be true. If it was, gases would never condense to form liquids. In reality, there is a small force of attraction between gas molecules that tends to hold the molecules together. This force of attraction has two consequences: (1) gases condense to form liquids at low temperatures and (2) the pressure of a real gas is sometimes smaller than expected for an ideal gas. To correct for the fact that the pressure of a real gas is smaller than expected from the ideal gas equation, van der Waals added a term to the pressure in this equation. This term contained a second constant (a) and has the form:

an2/V2

The complete van der Waals equation is therefore written as follows:

This equation is something of a mixed blessing. It provides a much better fit with the behavior of a real gas than the ideal gas equation. But it does this at the cost of a loss in generality. The ideal gas equation is equally valid for any gas, whereas the van der Waals equation contains a pair of constants (a and b) that change from gas to gas.

When two or more gases that do not chemically interact are found in one vessel, each gas will behave independently of the others. That is, each gas will behave as if it were the only gas in the container. Therefore, the pressure exerted by each gas in the mixture will be equal to the pressure that the gas would exert if it were the only one in the container. The pressure exerted by each individual gas is called the partial pressure of that gas. In 1801, John Dalton derived an expression, now known as Dalton’s law of partial pressures, which states that the total pressure of a gaseous mixture is equal to the sum of the partial pressures of the individual components. The equation for Dalton’s law is

PT = PA + PB + PC + …

where PT is the total pressure in the container, and PAPB, and PC are the partial pressures of gases A, B, and C, respectively. When more than one gas is in a container, each contributes to the whole as if it were the only gas present. Add up all of the pressures of the individual gases and you get the whole pressure of the system. The partial pressure of a gas is related to its mole fraction and can be determined using the following equation:

PA = XA+ PT

XA = moles of gas A / total moles of gas

The sum of the mole fractions in a system will always equal 1. The mole fraction (X in the above equation) is used to calculate the vapor pressure depression of a solution.

References

1) Bettini, Alessandro (2016). A Course in Classical Physics 2—Fluids and Thermodynamics. Springer. p. 8

2) The Venturi effect”. Wolfram Demonstrations Project. Retrieved 2017-08-06

3) Alberts, B.; Johnson, A.; Lewis, J.; Raff, M.; Roberts, K.; Walters, P. (2002). Molecular Biology of the Cell (4th ed.). New York and London: Garland Science.

4) Moran and Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, 4th Ed, 2000.

5) Kauzmann, W. (1966). Kinetic Theory of Gasses, W.A. Benjamin, New York

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