Gases and Gas Laws. Relationships of the Physical Properties of Gases. What is a gas?

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1 What is a gas? For pure substances, there are many different phases of matter, but only three that we interact with at common temperature and pressures solids, liquids and gases. We can differentiate the phases on three criteria: how the particles are paced, how they move and how they interact. The particles of solids and liquids are very tightly paced, with the exception of water, solids more so than liquids. Gases, however, have a great deal of space between particles, far greater than each individual particle. As an analogy picture a large and empty warehouse. As you and a friend enter the warehouse you may randomly move about at will, occasionally bumping into one another, but typically the only thing you would move into are the walls of the warehouse. Such is a gas. Now fill the warehouse with people so that you can no longer move about. Such tight pacing is characteristic of solids and liquids. The tight pacing allows for contact between individual particles, which in turn allows for attractive forces to hold the particles together. Because gas particles are very diffuse (lots of space in-between) they do not interact except for the occasional bumping into one another. One way we now this to be true is that we can easily pass between water particles as steam and not so easily through liquid water or ice. Within a substance, the greater the attraction, the more tightly the particles are held together, thereby limiting the individual movement of each particle. So much so that solid particles can only vibrate in place, while liquids being less densely paced may vibrate and rotate. These movements are called ranges of motion. Gas particles also vibrate and rotate, but because the space between the particles is so much larger than the particles themselves, they can also move across space translate. So a gas is a substance with relatively large empty space between particles that do not interact as they vibrate, rotate and translate across space. Kinetic Molecular Theory Kinetic molecular theory is a set of assumptions for gases. First although an individual gas particle has a finite volume that volume is amazingly small relative to the total volume available so the assumption is that a gas particle is a point particle with no volume. Again picture the empty warehouse. If the warehouse has 0,000 ft, then as you enter, how much area is there for you to move about in? Most people would say 0,000 ft, but truthfully we need to account for the space you occupy so it is closer to 9,999 ft, since you cannot move into space in which you already are. This is the same for gases. The air in an empty -L soda bottle has an available space of -L only if we assume that each air particle does not occupy any space. The great distance between the particles allows for our next assumptions. As the particles move through the space they travel in straight lines. Furthermore, considering the vast distance between them, we will assume that there are no attractive or repulsive forces between the particles. The last assumptions deal with the energy of the particles. First, to simplify everyday calculations, we will assume that no energy is transferred when gas particles collide (elastic collisions). Next we assume that the average inetic energy is directly proportional to the temperature. Because inetic energy is based on the velocity (speed), the temperature is related to speed. Therefore if the temperature of a gas increases (or decreases) the average speed of the gas particles also increases (or decreases). This and the previous assumptions will allow us to discuss the s between the pressure, volume, temperature and number of moles of a gas. elationships of the hysical roperties of Gases To build the s between the physical properties of gases, we need to define the physical properties of gases. At this point there are four that we need to discuss: pressure, volume, temperature and the number of moles. ressure, Volume, V ressure is the force per unit area. A good one-word definition is collisions because pressure represents the force of collisions between the gas particles and the walls of the container. The units of pressure are: atmospheres, atm; iloascals, a; millimeters of mercury, mmhg; and torr. The conversions values for these units are:.00 atm = 0.3 a = 760 mmhg = 760 torr Volume is the available space, and space will be the one-word definition. The units are cubic decimeters, dm 3 ; liters, L; cubic centimeters, cm 3 ; and milliliters, ml. The conversion values between those units are: dm 3 = L = 000 cm 3 = 000 ml

2 Temperature, T Temperature is the average inetic energy of the particles, and speed will be the one-word definition. The units are Kelvin, K; and degrees Celsius, C. The values for Celsius temperature are relative to the melting point and freezing point of water, however the Kelvin temperature scale is an absolute temperature scale relative to nothing. A temperature of absolute zero, 0 K, represents zero average inetic energy and is the coldest possible temperature. Only an absolute temperature scale will give us the proper s between the physical properties of the gases. Therefore only the Kelvin temperature scale is used. For example a doubling in the temperature from 0 C 40 C is only a small change in the inetic energy which is seen in the small change in the Kelvin temperature, 93 K 33 K. Since a thermometer typically measures the temperature in C, but we use K in the gas laws, we must convert between C and K by adding 73: K = C Number of moles, n The number of gas particles present in any system is a count of gas particles and therefore has the units of moles. The symbol in any mathematical equation for moles is n, for the number of moles. Gas Laws Charles Law: elating V and T at constant and n For a sample of gas we can measure the space, V, and speed, T, (V and T ) at a constant pressure,, and sample size, n. If the gas is heated, the speed is increased (a second measured temperature, T ), which results in an increased space (a second measured volume, V ) in order for the number of collisions,, to be constant with a constant number of particles, n. Jacques Charles, ca 800, found that the increase in the space, V, mirrored the initial increase in the speed, T, such that the direct proportion of V/T equaled a constant, : V T =. Since both sets of V and T equal, then the two sets equal each other: V T and V T so.... V V this equation is Charles Law. T T Gay-Lussac s Law: elating and T at constant V and n For a sample of gas we can measure the collisions,, and speed, T, ( and T ) at a constant volume, V, and sample size, n. If the speed is increased (a second measured temperature, T ) then the number of collisions must also increase (a second measured pressure, ) if the space confining the gas particles is unchanged (constant volume) with a constant number of particles, n. Joseph Gay-Lussac wored with gases and gas pressure in the early 800 s, and is credited with finding the between pressure and temperature for gases. His wor was based on the wor of Guillaume Amontons who studied the between temperature and pressure a century earlier. They discovered that if the speed of a gas is increased form T T then the number of collisions proportionally increased from. Such that the direct proportion of /T equaled a constant, : T =. Since both sets of and T equal, then the two sets equal each other: T and T so.... this equation is Gay-Lussac s Law. T T Boyle s Law: elating and V at constant T and n For a sample of gas we can measure the collisions,, and space, V, ( and V ) at a constant temperature, T, and sample size, n. If the space is increased (a second measured volume, V ) then the number of collisions will decrease (a second measured pressure, ) if the speed of the gas particles is unchanged (constant temperature) with a constant number of particles, n. obert Boyle woring in the 600 s studied the between the volume of a gas and the gas pressure. He discovered that if the volume of a gas is increased form V V then the number of collisions proportionally decreased from. Such that the indirect proportion of.v equaled a constant, : V =. Since both sets of and V equal, then the two sets equal each other: V. and V so.... V V this equation is Boyle s Law

3 Avogadro s Law: elating V and n at constant and T For a sample of gas we can measure the space, V, and the sample size, n, (V and n ) at a constant pressure,, and temperature, T. If additional gas is added to the system (a second sample size, n ) then for the both the speed and the number of collisions to remain unchanged then the space must also increase (a second measured volume, V ). These conclusions are the result of wor done by both Gay-Lusacc and Amadeo Avogadro in the early 800 s. They discovered that if the sample size of a gas is increased from n n then the volume (available space) is proportionally increased from V V. Such that the direct proportion of V/n equaled a constant, : V =. Since n both sets of V and n equal, then the two sets equal each other: V n and V n so.... V V this equation is Avogadro s Law. n n ecall from our study of mole conversions and stoichiometry that the coefficient ratio for a chemical reaction is the mole ratio. earrangement of Avogadro s law by cross-multiplication results in the equation: V n V n The volume ratio equals the mole ratio for gases. This is the most powerful result of this law and why Avogadro s law is often applied to chemical reactions. For example, if H (g) + O (g) H O(g), then the volume of H O(g) produced is equal to the volume of H used and twice the volume of O. Note that the volumes of the gases are not additive (+ 3) but instead are based on the mole ratio ::. Combined and Ideal Gas Law: elating, V, n, and T If we combine the previous results: V T =, T =, V =, and V n =, then the proportion becomes V V nt = where the constant is the gas constant and given a new symbol,, so that =. nt V nt V nt This equation is typically written without the fractions and becomes V = nt the Ideal Gas Law. This equation is typically used when only one set of, V, n, T data is present for a gas sample. Most students find that the equation is most easily memorized if you say it as a word pivnert. If, V, n, T data is measured for a gas sample and then that same gas is subjected to new conditions such that a second set of, V, n, T data is nown, then V nt = for both sets of data. If we combine the two sets we generate the equation: V V n T = V n T nt and the Combined Gas Law. V nt so.... V V this equation is Combined Gas Law. n T n T This equation is typically used when there are two sets of, V, n, T data present for a gas sample. Additionally, the other gas laws can be derived from the combined gas law. So when solving gas law problems with two sets of data we can start with the combined gas law, and then cancel out any variable that is held constant not mentioned. (see the example problems that follow) The advantage here is that if you now the ideal gas law, V = nt, then you can easily rearrange it into the combined gas law and from there derive all the others. A Note About Units Temperature must be in Kelvin. The previous s are only true for Kelvin temperature. For example V Charles Law shows that if the temperature is doubled then the volume will double: T = V T. Experimentally this is found to be true if the temperature starts at 80 K, (7 C) and doubles to 560 K (87 C) and not true if the temperature had doubled from 7 C to 34 C. Furthermore, a temperature of -0 C does not create the impossible volume of -0 L. Volume typically is in liters, L. The units must be the same on both sides of the equation. Since, as we shall soon see, the gas constant,, has liters, L, in its units, then the volume must be in liters as well.

4 ressure has a variety of units. ressure can be measured in atm, a, mmhg or in torr. The units must be the same on both sides of the equation. For the ideal gas law the units can either be converted to match the pressure unit in the gas constant,, or you can choose a value of that matches the given pressure unit. Gas Constant has a variety of units. The following values/units for the gas constant are all equivalent and may be used interchangeably in first year high school chemistry. The exception occurs in your advanced studies. If energy is involved inetic energy of gases, root mean square velocity, etc then only a is acceptable the value of 8.34 must be used. art of the reasoning is that the units must derive to the energy unit of joules, J. The values and units a L atm L mmhg L torr L for the gas constant are: Example of Gas Law roblems Ex mol K mol K mol K mol K Terms, Symbols and Equations for Gases pressure, force per unit area, (collisions) units: atm = 0.3 a = 760 torr = 760 mm Hg partial pressure, p pressure due to one individual gas in a mixture of gases volume, V available space, (space) units: dm 3 = L = 000 cm 3 = 000 ml temperature, T average inetic energy of all the particles in the system, (speed) unit: K = C +73 number of moles, n number of particles in the system, may be of one type of gas or of all gases mole fraction, percentage of a gas in a mixture expressed as a decimal diffusion the movement of gases from high concentration (high pressure) to areas of low concentration (low pressure). The movement of a gas is independent of other gases. gas constant, 8.34 V n T La molk Combined Gas Law use when there are two sets of data, cancel anything held V n T constant or not mentioned Ideal Gas Law V = nt use when there is only one set of,v,t data Dalton s Law total = p + p + p 3... the total pressure is the sum of the individual partial pressures x total = p i dt Density of Gases M = the density of a gas is directly related to the molar mass Molar Volume, V m Vm. 44 only at standard temperature and pressure Boyle s Law V = V at constant T and n; inverse V V T T Charles Law at constant V and n; direct T T Gay-Lusac s Law at constant and n; direct Latm molk Ltorr molk L mol LmmHg molk

5 n n V V Avogadro s Law at constant and T where n is the coef.; direct The Graphs: vs V (constant T & n) V vs T (constant & n) vs T (constant V & n) V V T T