Physical Chemistry I (Gases) Past Paper PDF 2016

Physical chemistry I (Gases) Dr Fateh Eltaboni KINETIC MOLECULAR THEORY OF GASES  The word kinetic means motion.  Maxwell and Boltzmann (1859) developed a mathematical theory to explain the behaviour of gases and the gas laws. Kinetic Theory is based on:  Gas is ma...

Physical chemistry I (Gases) Dr Fateh Eltaboni KINETIC MOLECULAR THEORY OF GASES  The word kinetic means motion.  Maxwell and Boltzmann (1859) developed a mathematical theory to explain the behaviour of gases and the gas laws. Kinetic Theory is based on:  Gas is made of a large number of molecules in continuous motion. ASSUMPTIONS OF THE KINETIC THEORY: (1) A gas consists of small particles called molecules dispersed in the container. (2) The actual volume of the molecules is negligible compared to the total volume of the gas. (3) The molecules of a given gas are identical and have the same mass (m). (4) Gas molecules are in constant random motion with high velocities. 1  Physical chemistry I (Gases) Dr Fateh Eltaboni (5) Gas molecules move in straight lines with uniform velocity and change direction on collision with other molecules or the walls of the container. (6) The distance between the molecules are very large so van der Waals attractive forces between them do not exist. Thus the gas molecules can move freely. (7) Gases collisions are elastic. Hence, there is no loss of the kinetic energy of a molecule during a collision. (8) The pressure of a gas is caused by the collisions of molecules on the walls of the container. (9) The average kinetic energy (1/2 mv 2) of the gas molecules is directly proportional to absolute temperature (Kelvin temperature). The average kinetic energy of molecules is the same at a given temperature. 2  Physical chemistry I (Gases) Dr Fateh Eltaboni Difference between Ideal Gas and Real Gases Ideal Gas (Virtual) Real Gases (O2, N2, H2) 1 Agree with the assumptions Do not agree of the kinetic theory of gases 2 Obeys the gas laws under all Obey the gas laws under conditions of temperature moderate conditions of and pressure. temperature and pressure: At very low temperature and very high pressure, the real gases show deviations from the ideal gas behaviour. 3 The actual volume of The actual volume of molecules molecules is negligible is considerable 4 No attractive forces between There is attractive forces molecules between molecules 5 Molecular collisions are Non elastic collisions elastic DERIVATION OF KINETIC GAS EQUATION  Let us consider a certain mass of gas enclosed in a cubic box at a fixed temperature.  Suppose that : The length of each side of the box = l cm The total number of gas molecules = n The mass of one molecule = m The velocity of a molecule = v 3  Physical chemistry I (Gases) Dr Fateh Eltaboni The kinetic gas equation derived by the following steps: (1) Random velocity of gas (ѵ) is a vector quantity and can be resolved into the components ѵ x, ѵ y, ѵ z along the X, Y and Z axes: Resolution of velocity v into components Vx , Vy and Vz. (2) The Number of Collisions Per Second on Face A Due to One Molecule:  Consider a molecule moving in (X) direction between opposite faces (A ) and (B). It will strike the face (A) with velocity (ѵ x) and rebound with velocity (– ѵ x).  To hit the same face again, the molecule must travel (1 cm) to collide with the opposite face (B) and then again (1 cm) to return to face A. 4  Physical chemistry I (Gases) Dr Fateh Eltaboni Cubic box showing molecular collisions along X axis. Therefore: (3) Each collision of the molecule on the face (A) causes a change of momentum (mass × velocity): Momentum before collision Momentum after collision Momentum change But the number of collisions per second on face (A) = Therefore, the total change of momentum per second on face A caused by one molecule: The change of momentum on both faces (A) and (B) along (X)- axis would be double: 5  Physical chemistry I (Gases) Dr Fateh Eltaboni Similarly, the change of momentum along (Y)-axis and (Z)-axis will be: Hence, the overall change of momentum per second on all faces of the box will be: (4) Total Change of Momentum Due to collisions of All the Molecules on All Faces of the Box: Suppose there are (N) molecules in the box each of which is moving with a different velocity (v1, v2, v3), etc. The total change of momentum due to collisions of all the molecules on all faces of the box: (5) Calculation of Pressure from Change of Momentum; Derivation of Kinetic Gas Equation 6  Physical chemistry I (Gases) Dr Fateh Eltaboni Since force may be defined as: the change in momentum per second, we can write:  This is the fundamental equation of the kinetic molecular theory of gases. It is called the Kinetic Gas equation. Root Mean Square (u) or (RMS) Velocity: KINETIC GAS EQUATION IN TERMS OF KINETIC ENERGY: 7  Physical chemistry I (Gases) Dr Fateh Eltaboni Where (E) is the total kinetic energy of all the N molecules. The equation (1) called the kinetic gas equation in terms of kinetic energy.  We know that the General ideal gas equation is:  Substituting the values of (R, T, N0), in the equation (5), the average kinetic energy of one gas molecule can be calculated. 8  Physical chemistry I (Gases) Dr Fateh Eltaboni DEDUCTION OF GAS LAWS FROM THE KINETIC GAS EQUATION: (a) Boyle’s Law According to the Kinetic Theory: T (K) α E 9  Physical chemistry I (Gases) Dr Fateh Eltaboni Therefore the product (PV) will have a constant value at a constant temperature. This is Boyle’s Law. (b) Charles’ Law As derived above: At constant pressure, volume of a gas is proportional to Kelvin temperature and this is Charles’ Law (c) Avogadro’s Law If equal volume of two gases be considered at the same pressure, When the temperature (T) of both the gases is the same, their mean kinetic energy per molecule will also be the same. Dividing (1) by (2), we have 10  Physical chemistry I (Gases) Dr Fateh Eltaboni Or, under the same conditions of temperature and pressure, equal volumes of the two gases contain the same number of molecules. This is Avogadro’s Law. (c) Graham’s Law of Diffusion If (m1) and (m2) are the masses and (u1) and (u2) the velocities of the molecules of gases 1 and 2, then at the same pressure and volume 11  Physical chemistry I (Gases) Dr Fateh Eltaboni DISTRIBUTION OF MOLECULAR VELOCITIES Maxwell’s Law of Distribution of Velocities dNc = number of molecules having velocities between C & (C + dc) N = total number of molecules, M = molecular mass T = temperature on absolute scale (K) Distribution of molecular velocities in nitrogen gas, N2, at 300 K and 600 K. KINDS OF VELOCITIES Three different kinds of molecular velocities: (1) Average velocity ( ) (2) Root Mean Square velocity (μ) (3) Most Probable velocity (vmp) 12  Physical chemistry I (Gases) Dr Fateh Eltaboni (1) Average Velocity ( ): For (n) molecules of a gas having individual velocities v1, v2, v3..... vn. The average velocity is the mean of the various velocities of the molecules: From Maxwell equation the average velocity can be calculated when Temperature (T) and Molar mass of gas molecule are known: (2) Root Mean Square Velocity (μ): If v1, v2, v3..... vn are the velocities of (n) molecules in a gas, (μ2), the mean of the squares of all the velocities is: Taking the root: (μ) is thus the Root Mean Square velocity or (RMS) velocity. 13  Physical chemistry I (Gases) Dr Fateh Eltaboni  The value of the (RMS) of velocity (μ), at a given temperature can be calculated from the Kinetic Gas Equation: For one mole of gas (n=1): Note: M = (kg/mol) (3) Most Probable Velocity (vmp):  The most probable velocity (vmp) is the velocity of the largest number of molecules in a gas.  According to the calculations made by Maxwell, the most probable velocity (vmp) is given by: vmp = 14  Physical chemistry I (Gases) Dr Fateh Eltaboni Relation between Average Velocity ( ), RMS Velocity (μ) and Most Probable Velocity (vmp) ( ) and (μ): Average Velocity = 0.9213 × RMS Velocity  (vmp) and (μ): Most Probable Velocity = 0.8165 × RMS Velocity ☺What is the relationship between ( ) and (vmp)???? 15  Physical chemistry I (Gases) Dr Fateh Eltaboni CALCULATION OF MOLECULAR VELOCITIES  The velocities of gas molecules are exceptionally high. Thus velocity of hydrogen molecule (H2) is 1,838 metres sec–1.  It may appear impossible to measure so high velocities, these can be easily calculated from the Kinetic Gas equation. (R = 8.314 J/mol K and M = kg/mol) Calculations of RMS: 16  Physical chemistry I (Gases) Dr Fateh Eltaboni Calculation of Molecular Velocity at STP: At STP: n = 1 mol, V= 22.4 L, P = 1 atm and T = 0 oC. 17  Physical chemistry I (Gases) Dr Fateh Eltaboni Calculation of most probable velocity: 18  Physical chemistry I (Gases) Dr Fateh Eltaboni TRANSPORT PROPERTIES  The derivation of Kinetic gas equation DID NOT take into account collisions between molecules.  The molecules in a gas are constantly colliding with one another.  The transport properties of gases such as diffusion, viscosity and mean free path depend on molecular collisions. The Mean Free Path():  At a given temperature, a molecule travels in a straight line before collision with another molecule.  The distance travelled by the molecule before collision is termed free path (l).  The mean distance travelled by a molecule between two collisions is called the Mean Free Path. It is denoted by (λ).  If l1, l2, l3 are the free paths for a molecule of a gas, its mean free path equals: 19  Physical chemistry I (Gases) Dr Fateh Eltaboni  Where (n ) is the number of molecules with which the molecule collides.  The number of molecular collisions will be less at a lower pressure or lower density AND the mean free path will be longer.  By a determination of the viscosity of the gas, the mean free path can be calculated. At STP, the mean free path for H2 is 1.78 × 10–5 cm and for O2 it is 1.0 × 10–5 cm. 20  Physical chemistry I (Gases) Dr Fateh Eltaboni Effect of Temperature on Mean Free Path: Effect of Pressure on Mean Free Path: We know that the pressure of a gas at certain temperature is directly proportional to the number of molecules per c.c. i.e.: 21  Physical chemistry I (Gases) Dr Fateh Eltaboni  Thus, the mean free path of a gas is directly proportional to inversely proportional to the pressure of a gas at constant temperature. COLLISION NUMBER: 22  Physical chemistry I (Gases) Dr Fateh Eltaboni The Collison Diameter (σ):  The closest distance between the centres of the two molecules taking part in a collision is called the Collision Diameter. It is denoted by (σ).  The smaller collision diameter of two molecules MEANS the larger mean free path. 23  Physical chemistry I (Gases) Dr Fateh Eltaboni The Collision Frequency (Z):  It's the number of molecular collisions taking place per second per unit volume (c.c.) of the gas.  Evidently, the collision frequency of a gas increases with increase in: 1. Temperature 2. Molecular size 3. Number of molecules per c.c. Effect of Temperature on Collision Frequency We know collision frequency is given by: From this equation it is clear that  Remember that: Hence collision frequency is directly proportional to the square root of absolute temperature. Effect of Pressure on Collision Frequency: From this equation ( ) we have: 24  Physical chemistry I (Gases) Dr Fateh Eltaboni Where is the number of molecules per c.c. But we know that: Thus Thus the collision frequency is directly proportional to the square of the pressure of the gas. 25  Physical chemistry I (Gases) Dr Fateh Eltaboni SPECIFIC HEAT RATIO OF GASES  The Specific heat (C): It is defined as the amount of heat required to raise the temperature of one gram of a substance (1°C). Specific Heat at Constant Volume (Cv):  It is the amount of heat required to raise the temperature of one gram of a gas (1°C) while the volume is kept constant and the pressure allowed to increase.  It is possible to calculate its value by making use of the Kinetic theory:  Assume that the heat supplied to a gas at constant volume is used entirely in increasing the kinetic energy of the moving molecules, and consequently increasing the temperature, Thus: 26  Physical chemistry I (Gases) Dr Fateh Eltaboni Specific Heat at Constant Pressure (Cp)  It is defined as the amount of heat required to raise the temperature of one gram of gas (1°C), the pressure is constant while the volume is allowed to increase.  When a gas is heated under constant pressure, the heat supplied is used in two ways : (1) In increasing the kinetic energy of the moving molecules: 3/2 R + x cal (2) The work done by the gas is equivalent to the product of the pressure and the change in volume. (∆V). For 1 g mole of the gas at temperature T, Hence (R) cal must be added to the value of (3/2 R) cal to get the thermal equivalent of the energy supplied to one gram of the gas in the form of heat when its temperature is raised by 1°C. 27  Physical chemistry I (Gases) Dr Fateh Eltaboni Specific Heat Ratio (ᵞ):  When experiment is done at 15°C : specific heat ratio helps us to determine the atomicity of gas molecules.  The theoretical difference between Cp and Cv is R and equals to about 2 calories.(Cp - Cv = R) 28  Physical chemistry I (Gases) Dr Fateh Eltaboni DEVIATIONS FROM IDEAL BEHAVIOUR  An ideal gas is one which obeys the gas laws or the gas equation (PV = RT) at all pressures and temperatures.  In fact no gas is ideal. Almost all gases show significant deviations from the ideal behaviour.  Thus the gases H2, N2 and CO2 which fail to obey the ideal- gas equation are termed non ideal or real gases. Compressibility Factor (Z): The amount to which a real gas deviates from the ideal behaviour is called Compressibility factor (Z). It is defined as:  The deviations from ideality may be shown by a plot of the compressibility factor (Z)(y-axis) against (P)(x-axis).  For an ideal gas (Z = 1) and it is independent of temperature and pressure.  The deviations from ideal behaviour of a real gas will be determined by the value of (Z) being greater or less than (1).  For a real gas (Z >1 or Z