kinetic theory and gas pressure

Kinetic Theory and Gas Pressure

Objectives

(d) define the term pressure and use the kinetic model to explain the pressure exerted by gases;

(i) state the basic assumptions of the kinetic theory of gases;

OutcomesALL MUST

Be able to define pressure.

Be able to use the kinetic model to describe pressure.

MOST SHOULD

Be able to state the basic assumptions of the kinetic theory of gases.

Be able to use the kinetic model to explain the pressure exerted by gases.

Be able to select and use the equation p=1/3ρv2

SOME COULD

Be able to derive the equation p=1/3ρv2

Pressure

P (Pa) = F (N) / A (m2)

Number ofmolecules

Speed (c)

cp the mostprobable speed

c the meanspeed (theaverage speedof all of themolecules)

crms the rms speedie the root meansquare speed. Auseful concept.Listen carefullyand watch how itis calculated.

The Kinetic Theory of Gases

Thermodynamics

Question (We do not have enough particles here to be realistic but it will illustrate the point!)

6 particles have the following speeds: 600, 650, 650, 700, 725, 750ms-1. Determine the most probable speed cp, the mean speed and the root mean square speed crms.


Thermodynamics

The most probable speed cp = 650ms-1 as there are more particles going at that speed than any other.

6 particles have the following speeds: 600, 650, 650, 700, 725, 750ms-1.


Thermodynamics


The mean speed c= (600 + 650 + 650 + 700 + 725 + 750) / 6 = 679.16ms-1.


Thermodynamics

The mean square speed= (6002 + 6502 + 6502 + 7002 + 7252 + 7502) / 6


= 2,783,125 / 6= 463854.17

The root mean square speed crms

= 46385417, .= 681.07ms-1

Assumptions of the kinetic theory of an IDEAL GAS.

1 1 A Gas consists of particles called molecules.

2 2 The molecules are in constant random motion. As many travelling in one direction as any other. The centre of mass

of the gas is at rest.

3 Intermolecular forces are negligible.

4 The duration of collisions between molecules is negligible.

5 Molecules move with constant velocity in between collisions.

6 6 The volume of gas molecules is negligible compared with the volume of the gas.

7 All collisions are totally elastic.

8 Newtonian mechanics can be applied to the collisions.


Thermodynamics

l

l

l

m,cx

x

y

z

If these assumptions are correct, we should be able to prove the equation of state for an ideal gas from these assumptions!

WOW!WOW!


Thermodynamics

l

l

l

m,cx

x

y

z

Consider the change in momentum as the particle hits the wallp = mcx - -mcx = 2mcx

Time interval between collisions t = 2l/cx

Now F=dp/dt from Newton’s second law so the force Fof one molecule hitting the wall is given by:F=p/t = 2mcx / 2l/cx = mcx

2 / l

But p = F/A so p = (mcx2 / l) / l2 = mcx

2 / l3

If there are N of them thenp = (m / l3 ) (cx1

2 + cx22 + cx3

2 +...........+ cxN2)

Note that (cx12 + cx22 + cx32 +...........+ cxN

2) / N = cx2

so p = (m / l3 ) Ncx2= N(m/V) cx

2 EQ(1)


Thermodynamics

so p = (m / l3 ) Ncx2= N(m/V) cx

2 EQ(1)

l

l

l

m,cx

x

y

zPythagoras’ Theorem will show thatc2 = cx

2 + cy2 + cz

2 - considering a general direction

and so c2 = cx2 + cy

2 + cz2

Due to the large no. of particles, cx2 = cy

2 = cz2

so c2 = 3 cx2

from EQ(1) we get pNm

Vc

1

32

pV Nmc1

32

So

OutcomesALL MUST

Be able to define pressure.

Be able to use the kinetic model to describe pressure.

MOST SHOULD

Be able to state the basic assumptions of the kinetic theory of gases.

Be able to use the kinetic model to explain the pressure exerted by gases.

Be able to select and use the equation p=1/3ρv2

SOME COULD

Be able to derive the equation p=1/3ρv2

kinetic theory and gas pressure

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