CHAPTER III: Kinetic Theory of Gases [5%]

Transcription

1 CHAPTER III: Kinetic Theory of Gases [5%] Introduction The kinetic theory of gases (also known as kinetic-molecular theory) is a law that explains the behavior of a hypothetical ideal gas. According to this theory, gases are made up of tiny particles in random, straight line motion. They move rapidly and continuously and make collisions with each other and the walls. This describes the gas pressure in terms of collisions with the walls of the container, rather than from static forces that push the molecules apart. Kinetic theory also explains how the different sizes of the particles in a gas can give them different, individual speeds. Kinetic theory makes the following assumptions. 1. Gases consist of particles in constant, random motion. They continue in a straight line until they collide with something usually each other or the walls of their container.. Particles are point masses with no volume. The particles are so small compared to the space between them, that we do not consider their size in ideal gases. 3. No molecular forces are at work. This means that there is no attraction or repulsion between the particles. 4. Gas pressure is due to the molecules colliding with the walls of the container. All of these collisions are perfectly elastic, meaning that there is no change in energy of either the particles or the wall upon collision. No energy is lost or gained from collisions. 5. The time it takes to collide is negligible compared with the time between collisions. 6. The kinetic energy of a gas is a measure of its Kelvin temperature. Individual gas molecules have different speeds, but the temperature and kinetic energy of the gas refer to the average of these speeds. 7. The average kinetic energy of a gas particle is directly proportional to the temperature. An increase in temperature increases the speed in which the gas molecules move. 8. All gases at a given temperature have the same average kinetic energy. 9. Lighter gas molecules move faster than heavier molecules. The mole and the Avogadro constant The unit for the amount of a substance is the mole. It is defined as the amount of a substance that contains the same number of particles (atoms or molecules) as the number of atoms in 1 g of the isotope carbon-1. 1mole = atoms This number is known as the Avogadro constant (N A) i.e. and is 6.0 x 10 3 particles per mole. N A = 6.0 x 10 3 per mole The ratio of the mass of one mole of the substance to one twelfth of the mass of one mole of carbon-1 is called the relative molecular mass of the substance - it is 3 for oxygen, for hydrogen and so on. A knowledge of the Avogadro constant enables us to calculate the number of molecules in any mass of a substance and therefore to get an idea of the size of one molecule. For example, a drop of water of volume 1.0 cm 3 has a mass of 1g. The relative molecular mass of water is 18, and therefore this drop of water must contain 6.0 x = 3.34 x 10 molecules. Avogadro's constant is the number of particles in a mole of the substance. This number is always the same. So in: 1 mole of hydrogen ( g) there are 6.0 x 10 3 molecules (hydrogen exists as H ) 1 mole of oxygen (3 g) there are 6.0 x 10 3 molecules (oxygen exists as O ) 1 mole of copper (63 g) there are 6.0 x 10 3 atoms 1 mole of uranium 35 (35 g) there are 6.0 x 10 3 atoms Prepared by Amit Dahal, YHSS Page 1 of 16

2 For example if we have kg of uranium in a fuel rod we have = 8.51 moles and this contains 8.51 x 6.0 x 10 3 = 5.1 x 10 3 atoms and so 5.1 x 10 3 uranium nuclei. The average volume of a water molecule must therefore be.99 x 10-3 cm 3. Molar mass and molecular mass Molar mass of a substance is the mass of one mole of the substance i.e. 6.0x10 3 molecules. The molecular mass is the mass of one molecule of the substance. Molecular mass = Molar mass Avogadro constant For example, the molecular mass of hydrogen is [/6.0x10 3 ] gm = 3.3 x 10-4 gm = 3.3 x 10-7 kg, and for oxygen [16/6.0 x 10 3 ] gm =.66 x 10-4 gm =.66 x 10-6 kg. STP Standard temperature and pressure, refers to nominal conditions in the atmosphere at sea level. Standard temperature is defined as zero degrees Celsius (0 0 C), which translates to 3 degrees Fahrenheit (3 0 F) or K. This is essentially the freezing point of pure water at sea level, in air at standard pressure. NTP Normal Temperature and Pressure - is defined as air at 0 0 C (93.15 K, 68 0 F) and 1 atm ( kn/m, kpa, 760 torr). Gas Laws Boyle's Law Boyle s law is a quantitative relationship between volume and pressure of a gas at constant temperature. It states that the volume of a given mass of a gas is inversely proportional to pressure if temperature remains constant. At constant temperature, the product of pressure and volume of a gas remains constant According to Boyle s law 1 P = constant 1 P P = K (constant) Let at pressure i. P 1, volume be 1 P 1 1 = constant. (i) ii. P, volume be P = constant.. (ii) Comparing (i) and (ii) P 1 1 = P Graph between & 1/P at constant temperature is a straight line. The value of k always stays the same so that P and vary appropriately. For example, if pressure increases, k must remain constant and thus volume will decrease. This is consistent with the predictions of Boyle's law. Prepared by Amit Dahal, YHSS Page of 16

3 Charles' Law It is quantitative relation between volume and absolute temperature of a gas at constant pressure. It states that the volume of a given mass of a gas at constant pressure is directly proportional to absolute temperature. The volume of a given mass of a gas increases or decreases by 1 /73 times of its original volume at 0 0 C for every degree fall or rise of temperature at given pressure. The ratio of volume to absolute temperature of a gas at given pressure is always constant. Let the volume of a gas at T Kelvin be Then according to Charles s law T = constant T = K (constant) T Let at temperature i. T 1 (kelvin), volume be 1 1 T 1 = K (constant). (i) ii. T (kelvin), volume be T = K (constant). (ii) Comparing (i) and (ii) 1 T 1 = T According to Charles' law, gases will expand when heated. The temperature of a gas is a measure of the average kinetic energy of the particles. As the kinetic energy increases, the particles will move faster and makes more collisions with the container. However, in order for the law to apply, the pressure must remain constant. The only way to do this is by increasing the volume. Let 0 be the volume of the given mass of a gas at 0 0 C, and t be the volume at t 0 C. Let T0 an T are the temperatures in Kelvin, then T 0 = 73+0 = 73 K and T = (73 + t) K According to Charle s law 1 T 1 = T 0 T 0 = t T 0 73 = t (73 + t) t = 0 ( (73 + t) 73 ) t = 0 (1 + t 73 ) Prepared by Amit Dahal, YHSS Page 3 of 16

4 What is an ideal gas? Since it's hard to exactly describe a real gas, the concept of an ideal gas allows us to model and predict the behavior of real gases. The term ideal gas refers to a hypothetical gas composed of molecules which follow a few rules: 1. Ideal gas molecules do not attract or repel each other. The only interaction between ideal gas molecules would be an elastic collision upon impact with each other or an elastic collision with the walls of the container.. Ideal gas molecules themselves take up no volume. The gas takes up volume since the molecules expand into a large region of space, but the Ideal gas molecules are approximated as point particles that have no volume of themselves. 3. Ideal gas follows all the gas laws perfectly. The pressure, P, volume, and temperature T of an ideal gas are related by a simple formula called the ideal gas law. P = nrt Where P is the pressure of the gas, is the volume taken up by the gas, T is the temperature of the gas, R is the gas constant, and n is the number of moles of the gas. According to Boyle s law, for a given mass of a gas at constant temperature, volume is related to pressure as 1 P (i) According to Charle s law, for a given mass of a gas at constant pressure, volume is related to temperature as T. (ii) Combining (i) and (ii) T P = Constant T P P T = constant For 1 mol of a gas, the constant has the same value for all gases and is called Universal Gas Constant denoted by R. So P = R P = RT T For n moles of gas, P = nrt ideal or perfect gas equation Boltzmann s constant (k) is a gas constant per molecule. It is the ratio of Universal constant to Avogadro s number. So it is represented as k = R R = kn N A A The numerical value of Boltzmann s factor k is 1.38 x 10-3 Jk -1 Number of moles is given by no. of molecules n = Avogadro snumber = N P = N. k. N N A N A. T P = NkT A Universal Gas Constant (R) R = P nt = pressure volume number of moles temperature = work done number of moles temperature So the universal gas constant represents the work done by a gas per mole per Kelvin. S.I unit of R is J mole -1 K -1 and its numerical value is 8.31 J mole -1 K -1. Prepared by Amit Dahal, YHSS Page 4 of 16

5 Pressure of ideal gas in terms of density ρ and molecular mass M of a gas From Ideal Gas equation P = nrt Density is ρ = m. (ρ = density; m = mass; = olume) Mass m is m = M n (M = molar mass; n = number of moles) Therefore density can be expressed as ρ = m M n = ρ M = n. (a) From Ideal gas equation P = nrt n = P RT (b) Comparing equations (a) and (b) ρ M = P RT P = ρrt M Real Gas and deviation of the behavior of real gas from ideal gas A gas which obeys the gas laws and the gas equation P = nrt strictly at all temperatures and pressures is said to be an ideal gas. The molecules of ideal gases are assumed to be volume less points with no attractive forces between one another. But no real gas strictly obeys the gas equation at all temperatures and pressures. Deviations from ideal behavior are observed particularly at high pressures or low temperatures. Consider a gas molecule (black square at the bottom left). olume of the gas molecule is insignificant/negligible in first figure. olume becomes significant in the second figure. Causes of Deviation from Ideal Behaviour The causes of deviations from ideal behaviour may be due to the following two assumptions of kinetic theory of gases. i. The volume occupied by gas molecules is negligibly small as compared to the volume occupied by the gas. ii. The forces of attraction between gas molecules are negligible. Prepared by Amit Dahal, YHSS Page 5 of 16

6 The first assumption is valid only at low pressures and high temperature, when the volume occupied by the gas molecules is negligible as compared to the total volume of the gas. But at low temperature or at high pressure, the volumes of molecules are no more negligible as compared to the total volume of the gas. The second assumption is not valid when the pressure is high and temperature is low. But at high pressure or low temperature when the total volume of gas is small, the forces of attraction become appreciable and cannot be ignored. The deviations from ideal gas behavior can also be illustrated as follows: The isotherms obtained by plotting pressure, P against volume, for real gases do not coincide with that of ideal gas, as shown below. It is clear from above graphs that the volume of real gas is more than or less than expected in certain cases. The deviation from ideal gas behaviour can also be expressed by compressibility factor, Z. Prepared by Amit Dahal, YHSS Page 6 of 16

7 Compressibility factor (Z): The deviation from ideal behaviour is expressed by introducing a factor Z known as compressibility factor in the ideal gas equation. The ratio of P to nrt is known as compressibility factor. The ratio of volume of real gas, real to the ideal volume of that gas, perfect calculated by ideal gas equation is known as compressibility factor. But from ideal gas equation: Therefore Z = P real nrt P perfect = nrt perfect/ideal = nrt P Z = P real nrt = real perfect/ideal For ideal or perfect gases, the compressibility factor, Z = 1. But for real gases, Z 1. Case-I: If Z>1 (Positive deviation) real > ideal The repulsion forces become more significant than the attractive forces. The gas cannot be compressed easily. Usually the Z > 1 for so called permanent gases like He, H etc. Case-II: If Z < 1 (Negative deviation) real < ideal The attractive forces are more significant than the repulsive forces. The gas can be liquefied easily. Usually the Z < 1 for gases like NH 3, CO, SO etc. The isotherms for one mole of different gases, plotted against the Z value and pressure, P at 0 o C are shown below: i. For gases like He, H the Z value increases with increase in pressure (positive deviation). It is because, the repulsive forces become more significant and the attractive forces become less dominant. Hence these gases are difficult to be condensed. Prepared by Amit Dahal, YHSS Page 7 of 16

8 ii. iii. For gases like CH 4, CO, NH 3 etc., the Z value decreases initially (negative deviation) but increases at higher pressures. It is because: at low pressures, the attraction forces are more dominant over the repulsion forces, whereas at higher pressures the repulsion forces become significant as the molecules approach closer to each other. But for all the gases, the Z value approaches one at very low pressures, indicating the ideal behavior. Also consider the following graphs of Z vs P for a particular gas, N at different temperatures. In above graphs, the curves are approaching the horizontal line with increase in the temperature i.e., the gases approach ideal behaviour at higher temperatures. Dalton s Law of Partial Pressure Partial Pressure In a mixture of different gases which do not react chemically, each gas behaves independently of the other gases and exerts its own pressure. This individual pressure that a gas exerts in a mixture of gases is called its partial pressure. Dalton s Law of Partial Pressure Based on this behavior of gases, JOHN DALTON formulated a basic law which is known as "The Dalton's law of partial pressure". "If two or more gases (which do not react with each other) are enclosed in a vessel, the total pressure exerted by them is equal to the sum of their partial pressure". Consider a mixture of three non-reacting gases a, b and c. Partial pressures of these gases are P a, P b and P c. According to Dalton's law of partial pressure, their total pressure is given by P total = P a + P b + P c According to kinetic molecular theory of gases there is no force of attraction or repulsion among the gas molecules. Thus each gas behaves independently in a mixture and exerts its own pressure. In terms of Kinetic molecular theory, Dalton's law of partial pressure can be explained as: "In a non-reacting mixture of gases, each gas exerts separate pressure on the container in which it is confined due to collision of its molecules with the walls of container. The total pressure exerted by the gaseous mixture is equal to the sum of collisions of the molecules of individual gas." Prepared by Amit Dahal, YHSS Page 8 of 16

9 Expression for Partial Pressure Consider a gaseous mixture of three different gases a, b and c enclosed in a container of volume dm 3 at T Kelvin. Let the partial pressures of these gases are P a, P b and P c respectively and total pressure of mixture is P t. Let there are n a, n b and n c moles of each gas respectively and the total number of moles are n t. Three gases confined in a cylinder under similar conditions: Using equation of state of gas: P = nrt OR P = nrt For gas a P a = n art (i) For gas b P b = n brt (ii) For gas c P c = n crt (iii) For any gas OR P gas = n gasrt P gas n gas = RT.. (a) Adding equation (i), (ii) and (iii), we get, But P t = n art + n brt + n crt = (n a + n b + n c ) RT n t = n a + n b + n c RT P t = n t P t = RT n t (b) Comparing equation (a) and (b), we get, P t = P gas n t n gas P gas = n gas P total n total This expression indicates that the pressure of a gas is proportional to number of moles if confined under similar conditions. Prepared by Amit Dahal, YHSS Page 9 of 16

10 Pressure of an Ideal Gas Z Consider an ideal gas (consisting of N molecules each of mass m) enclosed in a cubical box of side L. It s any molecule moves with velocity v in any direction where v v iˆ v x y ˆj v kˆ This molecule collides with the shaded wall and rebounds with velocity v x. z ( A1 ) with velocity The change in momentum of the molecule P ( mv ) ( mv ) mv x x x As the momentum remains conserved in a collision, the change in momentum of the wall A 1 is P mv After rebound this molecule travel toward opposite wall A with velocity with velocity v x towards wall A 1. x (1) Time between two successive collision with the wall A 1. Distance travelled by molecule between two successive collision t elocity of molecule v x v x Y v x L v x, collide to it and again rebound L v x 1 v x Number of collision per second n t L vx m () The momentum imparted per unit time to the wall by this molecule n P mv x vx L L m This is also equal to the force exerted on the wall due to this molecule F v x L m (3) The total force on the wall due to all the molecules F x v x L (4) Now pressure is defined as force per unit area Fx m m m m P x v x v x Similarly P y A AL v y and P z v z m So P x Py Pz ( v x v y v z ) m 3P v [As Px Py Pz P and v v x v y v z ] m 3 3P ( v1 v v3...) or or 3P mn v 1 3 vrms v v A 1 3 N v 4... A 1 P m N v As root mean square velocity of the gas molecule 1 v v 3 v 4... v rms N 1 m N or P v rms 3 Now the total mass of the gas M = mn, and since ρ = M we can write X Prepared by Amit Dahal, YHSS Page 10 of 16

11 1 P v rms 3 Kinetic Interpretation of gases According to the kinetic equation of pressure of a gas: 1 P v rms 3 But ρ = density of gas ρ = density of gas = mass of gas volume of gas ρ = density of gas = (M) 1 M P v rms..m: mass; : olume 3 Multiplying and dividing by 1 P Mv rms 3 1 Mv rms P K. E avg 3 For 1 mole of a gas P RT K. E avg RT 3 3 K. E avg RT 1 3 Mv rms RT is the average kinetic energy of the gas 3RT rms M Translational kinetic energy is the kinetic energy an object has due to its motion in a straight line from one place to another place. If N A is the Avagadro s number, then the mean kinetic energy per molecule is given by K. Erms 3 R K. Eavg T N N A 3 K. E avg kt k: Boltzmann s constant A rms Faster the motion of the molecules, higher will be the kinetic energy and hence higher will be the temperature of the gas. Hence the temperature of a gas is the measure of the average kinetic energy of its molecules. At T = 0, rms = 0 i.e. at absolute zero all the molecular motion stops. T Prepared by Amit Dahal, YHSS Page 11 of 16

12 Types of Molecular elocities The three types of molecular velocities are i. Most probable velocity ii. Average velocity iii. Root mean square velocity Most probable velocity (p) This is defined as the velocity possessed by maximum fraction of the total number of molecules of the gas at a given temperature. Where RT kt P p M M M R: gas constant T: temperature and M: molecular weight Average velocity ( avg) Average velocity is the mean of the velocities of all the molecules. It is calculated using the formula. avg 8RT 8kT 8P M M M Root mean square velocity It is defined as the square root of the mean of the square of the velocities of all the molecules of the gas. 3RT rms M 3kT 3P M M Root mean square velocity = x Average velocity Boltzmann distribution It is a distribution of velocities of gas particles in a particular temperature. It is obtained by plotting a graph between fraction (Δn/N) of molecules having different speeds against the speeds of the molecules (along x-axis). This curve is known as Maxwell's Boltzmann distribution curve. Even if the air is at a single temperature, the air molecules travels at different speed. Some of the air molecules will be moving extremely fast, some will be moving with moderate speeds, and some of the air molecules will hardly be moving at all. So rather than talking about the speed of an air molecule in a gas, we discuss the distribution of speeds in a gas at a certain temperature. The Maxwell-Boltzmann distribution is often represented with the graph as shown alongside. Prepared by Amit Dahal, YHSS Page 1 of 16

13 The y-axis of the Maxwell-Boltzmann graph can be thought of as giving the number of molecules per unit speed. So, if the graph is higher in a given region, it means that there are more gas molecules moving with those speeds. We notice that the graph is not symmetrical. There is a longer "tail" on the high speed right end of the graph. The graph continues to the right to extremely large speeds, but to the left the graph must end at zero (since a molecule can't have a speed less than zero). The average speed avg of a molecule in the gas is actually located a bit to the right of the peak. The reason the average speed is located to the right of the peak is due to the longer "tail" on the right side of the Maxwell-Boltzmann distribution graph. This longer tail pulls the average speed slightly to the right of the peak of the graph. Root-mean-square speed rms. is the square root of the mean of the squares of the velocities We can write the root-mean-square speed mathematically as, rms = ( n ) N Area under a Maxwell-Boltzmann distribution The y-axis of the Maxwell-Boltzmann distribution graph gives the number of molecules per unit speed. The total area under the entire curve is equal to the total number of molecules in the gas. If we heat the gas to a higher temperature, the peak of the graph will shift to the right (since the average molecular speed will increase). As the graph shifts to the right, the height of the graph has to decrease in order to maintain the same total area under the curve. Similarly, as a gas cools to a lower temperature, the peak of the graph shifts to the left. As the graph shifts to the left, the height of the graph has to increase in order to maintain the same area under the curve. This can be seen in the curves below which represent a sample of gas (with a constant amount of molecules) at different temperatures. As the gas gets colder, the graph becomes taller and narrower. Similarly, as the gas gets hotter the graph becomes shorter and wider. This is required for the area under the curve (e.g. total number of molecules) to stay constant. If molecules enter the sample, the total area under the curve would increase. Similarly, if molecules were to leave the sample, the total area under the curve would decrease. Prepared by Amit Dahal, YHSS Page 13 of 16

14 Kinetic Energy Distribution Maxwell and Boltzmann distribution helps us to examine the properties of very large numbers of molecules. While we can't predict the behavior of any one molecule with any precision, we can quite exactly predict the behavior of large quantities of molecules. Kinetic energy of an atom or molecule is calculated by the formula: KE = 1/ mv where m is the mass of the particle and v is its velocity. When we have to describe the distribution of kinetic energies in a large collection of atoms or molecules, the distribution can be described by plotting the graph as follows Notice that the plot is not symmetrical so the most probable speed is not the same as the average speed. Effect of temperature Temperature is directly proportional to the average kinetic energy of molecules, KE = 3 nrt where n is the number of moles of gas, T is the temperature in Kelvins, and R is the Universal gas constant If T is proportional to the average kinetic energy, then we'd expect for this change in the average kinetic energy to be reflected in changes in the shape of the Boltzmann distribution. As temperature increases, the curve will spread to the right and the value of the most probable kinetic energy will decrease. This is illustrated below for several temperatures. What happens to the fraction of molecules with kinetic energies equal to or greater than this value as we change the temperature? As T increases, the fraction of molecules with energies greater than the red line increase. This fact is of fundamental importance to properties of matter such as vapor pressures. Prepared by Amit Dahal, YHSS Page 14 of 16

15 Boltzmann Factor Equation Particles in a gas lose and gain energy at random due to collisions with each other. On average, over a large number of particles, the proportion of particles which have at least a certain amount of energy ε is constant. This is known as the Boltzmann factor. It is a value between 0 and 1. The Boltzmann factor is given by the formula: n = e ε kt n 0 where n is the number of particles with kinetic energy above an energy level ε, n 0 is the total number of particles in the gas, T is the temperature of the gas (in kelvin) and k is the Boltzmann constant (1.38 x 10 3 JK 1 ). This energy could be any sort of energy that a particle can have - it could be gravitational potential energy, or kinetic energy, for example. Derivation In the atmosphere, particles are pulled downwards by gravity. They gain and lose gravitational potential energy (mgh) due to collisions with each other. Considering a small portion of the atmosphere. Let horizontal cross-sectional area be A, height be dh, molecular density be n molecular mass be m. number of particles in the portion be N. n = N = N A dh By definition: N = n = na dh The total mass Σm is the mass of one molecule (m) multiplied by the number of molecules (N): Σm = mn = mna dh Then the weight of the portion of atmosphere: W = mass acc. due to fravity = gσm = mnga dh The downwards pressure P is force per. unit area, so: P = W nmga dh = = mngdh A A We know that, as we go up in the atmosphere, the pressure decreases. So, across the portion there is a difference in pressure dp given by: dp = nmgdh (1) In other words, the pressure is decreasing (-) and it is the result of the weight of this little chunk of atmosphere. We also know that: P = NkT So: P = NkT But we know that n = N Prepared by Amit Dahal, YHSS Page 15 of 16

16 So, by substitution P = nkt So, for portion of atmosphere dp = kt dn () Equating (1) and () dp = nmgdh = kt dn Rearranging, we get dn dh = nmg kt dn dh = kt nmg Integrating between the limits n 0 and n: h = kt n mg = 1 dn = kt n 0 n mg [ln n] n n 0 = kt mg (ln n ln n 0) = kt mg ln n n 0 ln n = mgh n 0 kt n = e mgh kt n 0 Since mgh is gravitational potential energy, ε = mgh, so n = e ε kt n 0 Prepared by Amit Dahal, YHSS Page 16 of 16