Ideal Gas Behavior. NC State University

Transcription

1 Chemistry 431 Lecture 1 Ideal Gas Behavior NC State University

2 Macroscopic variables P, T Pressure is a force per unit area (P= F/A) The force arises from the change in momentum as particles hit an object and change direction. Temperature derives from molecular motion (3/2RT = 1/2M<u 2 >) Mi l Greater average velocity results in a higher temperature. M is molar mass u is the velocity

3 Mass and molar mass We can multiply the equation: 3 2 RT = 1 2 M <u2 > by the number of moles, n, to obtain: 3 2 nrt = 1 2 nm <u2 > If m is the mass and M is the molar of a particle then we can also write: nm = Nm (N is the number of particles)

4 Mass and molar mass In other words nn A = N where N A is Avagadro s number nrt = 1 2 Nm <u2 > Average properties <u 2 > represents the average speed

5 Kinetic Model of Gases Assumptions: 1. A gas consists of molecules that move randomly. 2. The size of the molecules l is negligible. ibl 3. There are no interactions between the gas molecules. Because there are such large numbers of gas molecules in any system we will interested in average quantities. We have written average with an angle bracket. For example, the average speed is: <u 2 >=c 2 = s s s s N 2 N c = s 1 + s 2 + s s N N We use s for speed and c for mean speed.

6 Velocity and Speed When we considered the derivation of pressure using a kinetic model we used the fact that the gas exchanges momentum with the wall of the container. Therefore, the vector (directional) quantity velocity was appropriate. However, in the energy expression the velocity enters as the square and so the sign of the velocity does not matter. In essence it is the average speed that is relevant for the energy. Another way to say this is the energy is a scalar E = 1 2 m<u2 >= 1 2 mv2 = 1 2 mc2 p = mu = mv All of these notations mean the same thing.

7 The root-mean-square speed The ideal gas equation of state is consistent with an interpretation of temperature as proportional p to the kinetic energy of a gas M u = RT If we solve for <u 2 > we have the mean-square speed. u 2 = 3RT M If we take the square root of both sides we have the r.m.s. speed. 2 1/2 3RT u 2 = 3RT M

8 The mean speed The mean value is more commonly used than the root-mean-square of a value. The root-mean-square speed Is equal to the root-mean-square velocity: The mean speed is: c 2 = u 2 c = 8 c 2 3π The r.m.s. speed of oxygen at 25 o C (298 K) is 482 m/s. Note: M is converted to kg/mol! u 2 1/2 = J/mol K 298 K kg/mol = m / s

9 The Maxwell Distribution Not all molecules have the same speed. Maxwell assumed that the distribution of speeds was Gaussian. F(s) = 4π M 3/2 s 2 exp Ms 2 2πRT RT As temperature increases the r.m.s. speed increases and the width of the distribution increases. Moreover, the functions is a normalized distribution. This just means that the integral over the distribution function is equal to 1. 0 F(s) ds = 1 See the MAPLE worksheets for examples.

10 Cross section σ = πd 2 Molecular Collisions Center location of target molecule mean free path estimate = Interation Volume πd 2 <u>t <u>t distance traveled volume of interaction * number density n/v = moles per unit volume (molar density) N/V = molecules per unit volume (number density) mean free path estimate = < u > t σ < u > t N / V

11 Refinement of mean free path The analysis of molecular collisions assumed that the target atom was stationary. If we include the fact that the target atom is moving we find that t the relative velocity is: < u > rel = 2 < u > Therefore λ = < u > t 2σ < u > t N / V = 1 2σ N / V = 1 2σ N A n/v = A RT 2σ N A P A As the pressure increases the number density increases and the distance between collision (mean free path) becomes shorter. As the temperature increases at constant pressure the As the temperature increases at constant pressure the number density must decrease and the mean free path ill increase.

12 Mean free path Collision frequency The mean free path, λ is the average distance that a molecule travels between collisions. The collision frequency, z is the average rate of collisions made by one molecule. l The collision cross section, σ is target area presented by one molecule to another. When interpreted in the kinetic model it can be shown that: RT 2N Aσ u 2 P 2 λ = 2NA σp, z =, σ = πd RT The product of the mean free path and collision frequency is equal lto the room mean square speed. u 2 = λz

13 Units of Pressure Force has units of Newtons F = ma (kg m/s 2 ) Pressure has units of Newtons/meter 2 P= F/A = (kg m/s 2 /m 2 = kg/s 2 /m) These units are also called Pascals (Pa). 1 bar = 10 5 Pa = 10 5 N/m 2. 1 atm = x 10 5 Pa

14 Units of Energy Energy has units of Joules 1 J = 1 Nm Work and energy have the same units. Work is defined as the result of a force acting through a distance. We can also define chemical energy in terms of the energy per mole. 1 kj/mol 1 kcal/mol l = kj/mol

15 Thermal Energy Thermal energy can be defined as RT. Its magnitude depends d on temperature. R = 8.31 J/mol-K or 1.98 cal/mol-k At 298 K, RT = 2476 J/mol (2.476 kj/mol) Thermal energy can also be expressed on a per molecule basis. The molecular equivalent of R is the Boltzmann constant, k. R = N A k N A = x molecules/mol l l

16 Extensive and Intensive Variables Extensive variables are proportional to the size of the system. Extensive variables: volume, mass, energy Intensive variables do not depend on the size of the system. Intensive variables: pressure, temperature, t density

17 Equation of state relates P, V and T The ideal gas equation of state is PV = nrt An equation of state relates macroscopic properties which result from the average behavior of a large number of particles. P Macroscopic Microscopic

18 Microsopic view of momentum c u x b area = bc a A particle with velocity u x strikes a wall. The momentum of the particle changes from mu The momentum of the particle changes from mu x to mu x. The momentum change is Δp = 2mu x.

19 Transit time c u x b area = bc The time between collisions with one wall is Δt = 2a/u x. This is also the round trip time. a

20 Transit time c Round trip distance is 2a u x b area = bc a The time between collision is Δt = 2a/u x. velocity = distance/time. time = distance/velocity.

21 The pressure on the wall force = rate of change of momentum F = Δp Δt = 2mu x = mu 2 x 2a/u x a The pressure is the force per unit area. The area is A = bc and the volume of the box is V = abc P = F = mu x 2 = mu 2 x bc abc V

22 Average propertiesp Pressure does not result from a single particle striking the wall but from many particles. Thus, the velocity is the average velocity times the number of particles. 2 P = Nm u x V PV = Nm u2 x

23 Average propertiesp There are three dimensions so the velocity along the x-direction is 1/3 the total. 2 u x 3 = 1 3 u2 2 PV = Nm u 3 From the kinetic theory of gases 1 u Nm u2 = nrt 2

24 Putting the results together When we combine of microscopic view of pressure with the kinetic theory of gases result we find the ideal gas law. PV = nrt This approach assumes that the molecules have no size (take up no space) and that they have no interactions.