icroscopic Properties of Gases So far we he seen the gas laws. These came from observations. In this section we want to look at a theory that explains the gas laws: The kinetic theory of gases or The kinetic moleclar theory CHE 000.0 Gases icroscopic Kinetic Theory of Gases: ssmptions. gas is made p of a large nmber of extremely small particles (molecles or atoms) in constant, random, straight line motion. olecles occpy very little volme (most of the container is free space). olecles collide with one another and with the s of the container 4. There are no forces between the molecles 5. olecles can gain or lose energy on collision bt the total energy remains constant CHE 000.0 Gases icroscopic Collision between molecles Collision with CHE 000.0 Gases icroscopic
The theory will give information on the speeds of molecles, the freqency with which they collide, and the distribtion of energy It is only sefl if it can predict the gas laws CHE 000.0 Gases icroscopic 4 Pressre comes from the gas molecles hitting the s of the container. Hence if we can determine the force with which the molecles hit the we can determine the pressre. Sppose yo he a gas with identical molecles of mass m in a container of volme. lso assme that each molecle has a speed, bt that it can be different for different molecles. CHE 000.0 Gases icroscopic 5 y calclating the force that a molecle exerts on the, the nmber of collisions and eraging the reslt over the different moleclar speeds, one gets: P m where the erage of the sqares of the speeds. CHE 000.0 Gases icroscopic 6
P m This is looking P m a lot like P constant If is a constant at constant temperatre. This is oyle's Law We will not prove this bt assme it is tre and se the P eqations from the macroscopic and microscopic sections to learn abot speed and temperatre CHE 000.0 Gases icroscopic 7 Temperatre/kinetic energy/speed The kinetic energy of a molecle is: ½m The erage kinetic energy of a molecle is: m The kinetic energy of a mole of molecles is m ( is vogadro' s nmber) CHE 000.0 Gases icroscopic 8 The kinetic energy of a mole of molecles (E) is E m m E t P nrt given that m P nrt n.e.e E RT n.e CHE 000.0 Gases icroscopic 9
E RT This shows that the temperatre of the gas is a measre of the kinetic energy of the molecles CHE 000.0 Gases icroscopic 0 Eqating E Gives oleclar Speeds m E m RT RT RT m and E RT The root mean sqare speed rms RT CHE 000.0 Gases icroscopic What is the speed of a hydrogen molecle at 5 o C? RT 8.4 98 - rms.9 0 ms.06 0 (7,000km/hr) The factor of 0 - in the denominator is to pt the molar mass in SI nits CHE 000.0 Gases icroscopic 4
Distribtion of moleclar speeds So far we he calclated the root mean sqare speed only What other speeds are possible? What is the real erage speed? CHE 000.0 Gases icroscopic Distribtion of moleclar speeds m is the most probable speed (mode) is the erage speed (mean) rms is the root mean sqare speed ll speeds are possible bt are not eqally likely CHE 000.0 Gases icroscopic 4 verage speeds There are three erage speeds. Since the speed distribtion is not symmetric, they are different m rms RT 8RT π RT kt m 8kT πm kt m m : : rms 0.8 : 0.9 : CHE 000.0 Gases icroscopic 5 5
ariation in speed with temperatre We already know that the erage speeds vary with pt The red crves are for O at two temperatres. CHE 000.0 Gases icroscopic 6 How broad is the speed distribtion We know the erage speeds. Do most molecles he speeds similar to the erage? Do a lot go faster? CHE 000.0 Gases icroscopic 7 m m m Clearly the distribtion is narrow CHE 000.0 Gases icroscopic 8 6
It is possible to calclate the fraction of molecles that he a speed greater than Speed 0 m 5 m 0 m Fraction of molecles with speeds greater than (all molecles go faster than 0) 4.5x0-4 4x0-0 4x0-4 ost molecles he speeds close to m CHE 000.0 Gases icroscopic 9 Diffsion and Effsion Diffsion Effsion CHE 000.0 Gases icroscopic 0 Collisions with the Wall The nmber of collisions that molecles make with the depend on how many molecles there are (per nit volme) and how fast they are moving. Z will he nits of m - s - Z. actally Z 4. CHE 000.0 Gases icroscopic 7
Effsion We can obtain a qantitative nderstanding of effsion by recognizing that effsion is the loss of a molecle that wold normally hit the. Z. 4 The rate at which molecles lee the container is the collision rate times the area of the hole Rate of effsion Z. 4. CHE 000.0 Gases icroscopic. for rate of effsion of Z rate of effsion of Z gases () () ( and ) in the 8RT 8RT.() (). 4 ()..() 4 π π same container ote that the ratio of the nmber of molecles is the ratio of the partial pressres rate of effsion of P rate of effsion of P and for EQUL pressres rate of effsion of rate of effsion of CHE 000.0 Gases icroscopic Graham's Law.. side Stdents often he problems with problems involving rates and or time, becase they are inverses lways remember that Rate molecles time So the faster the rate the smaller the time CHE 000.0 Gases icroscopic 4 8
Collisions between molecles When stdying interactions between gas molecles yo need to know how often the molecles actally collide The collision freqency is the nmber of collisions a particlar molecle makes in one second. This will depend on how many molecles there are (per nit volme), how fast they are moving, and how big they are. CHE 000.0 Gases icroscopic 5 Collisions between molecles For a gas where all the molecles are the same, and he a diameter d, the collision freqency is Z P Z πd πd kt Z will he nits of s - CHE 000.0 Gases icroscopic 6 Collisions between molecles To calclate the total nmber of collisions for all molecles we mst mltiply Z by / to cont all the molecles, and by ½ so we don t cont them twice. Z πd Z will he nits of s - m - πd CHE 000.0 Gases icroscopic 7 9
ean free path related parameter is the erage distance a molecle trels between collisions. This is the mean free path The collision freqency is Z s - The time between collisions is /Z s The erage speed of the molecles is Since distance is speed times time λ Z πd CHE 000.0 Gases icroscopic 8 So how big are these nmbers?? itrogen at 98K 8RT π 8 8.4 98-475 m s - π 8.0 0 5 P.0 0 5.46 0 m kt.8 0 98 - d 4.0 0-0 m Z 475.46 0 4 4.9 0 - m s 5 7 - Z πd π ( 4.0 0 ) 475.46 0-0 475 λ Z 8. 0 5.6 0 56 nm 8 9 5 9 8. 0 s - CHE 000.0 Gases icroscopic 9 0