Collisions between molecules · randomly and undergoing millions of collisions every second. Shown is the distribution of molecular speeds in a sample of nitrogen gas at 20ºC. The

Collisions between molecules

• We model molecules as

rigid spheres of radius r as

shown at the right.

• The mean free path of a

molecule is the average

distance it travels between

collisions.

• The average time between

collisions is the mean free time.

• In a time dt a molecule with radius r will collide with any

other molecule within a cylindrical volume of radius 2r and

length v dt.

© 2016 Pearson Education Inc.

Collisions between molecules

• The average distance traveled between collisions is called the

mean free path.

• In our model, this is just the molecule’s speed v multiplied by

mean free time:

• The more molecules there are and the larger the molecule, the

shorter the mean distance between collisions.


© 2017 Pearson Education, Inc. Slide 20-3

The temperature of a rigid container of oxygen gas

(O2) is lowered from 300ºC to 0ºC. As a result, the

mean free path of oxygen molecules

A. Increases.

B. Is unchanged.

C. Decreases.

QuickCheck


Example 1

What is the mean free path of a nitrogen molecule at 1.0 atm

pressure and room temperature (20ºC)? Also, calculate the

mean time between collisions.


In-class Activity #1

What is the mean time between collisions for a monatomic

gas with a molar mass of M = 40.0 g/mol at STP? Assume

the diameter of the molecules to be 4.00 x 10-10 m.


Atoms in a monatomic gas carry energy exclusively as

translational kinetic energy (3 degrees of freedom).

Molecules in a gas may have additional modes of energy

storage, for example, the kinetic and potential energy

associated with vibration, or rotational kinetic energy.

We define the number of degrees of freedom as the

number of distinct and independent modes of energy

storage:

The Equipartition Theorem


A mass on a spring

oscillates back and forth on

a frictionless surface. How

many degrees of freedom

does this system have?

A. 1

B. 2

C. 3

D. 4

QuickCheck


The thermal energy of a

system is Eth = Kmicro + Umicro.

The figure shows a monatomic

gas such as helium or neon.

The atoms in a monatomic

gas have no molecular bonds

with their neighbors, hence

Umicro = 0.

Since the average kinetic energy of a single atom in an

ideal gas is Ktr = 3/2 kBT, the total thermal energy is

Thermal Energy and Specific Heat


If the temperature of a monatomic gas changes by

ΔT, its thermal energy changes by

Earlier we found that the change in thermal energy

for any ideal-gas process is related to the molar

specific heat at constant volume by

Combining these equations gives us a prediction for

the molar specific heat for a monatomic gas:

This prediction is confirmed by experiments.

Thermal Energy and Specific Heat


In addition to the 3 degrees of

freedom from translational kinetic

energy, a diatomic gas at

commonly used temperatures has

2 additional degrees of freedom

from end-over-end rotations.

This gives 5 degrees of freedom

total:

Diatomic Molecules


A. the monatomic gas

B. the diatomic gas

C. Both will undergo the same temperature change.

D. The answer depends on whether or not the diatomic

molecules can also vibrate.

QuickCheck

You have 1.00 mol of an ideal monatomic gas and 1.00 mol of an

ideal diatomic gas whose molecules can rotate. Initially both gases

are at room temperature. If the same amount of heat flows into each

gas, which gas will undergo the greatest increase in temperature?

Compare theory with experiment

• Table below shows that the calculated values for CV for

monatomic gases and diatomic gases agree quite well with

the measured values.


Experimental values of CV for hydrogen gas (H2)


© 2016 Pearson Education, Inc.

QuickCheck

The molar heat capacity at constant volume of diatomic hydrogen

gas (H2) is 5R/2 at 500 K but only 3R/2 at 50 K. Why is this?

A. At 500 K the molecules can vibrate, while at 50 K they cannot.

B. At 500 K the molecules cannot vibrate, while at 50 K they can.

C. At 500 K the molecules can rotate, while at 50 K they cannot.

D. At 500 K the molecules cannot rotate, while at 50 K they can.


Example 2

How much heat does it take to increase the temperature of

1.80 mol of an ideal gas by 50.0 K near room temperature

if the gas is held at constant volume and is diatomic? How

does the answer change is the gas is monatomic?


Example 3

Calculate the specific heat capacity at constant volume of water vapor,

assuming the nonlinear triatomic molecule has three translational and

three rotational degrees of freedom. The molar mass of water is 18.0

g/mol. Compare to the actual specific heat capacity of water vapor at

low pressures which is about 2000 J / kg K and consider whether

vibrational motion needs to be considered.



Perfectly rigid containers each hold n moles of ideal gas, one

being hydrogen (H2) and other being neon (Ne). If it takes

300 J of heat to increase the temperature of the hydrogen by

3.50ºC, by how many degrees will the same amount of heat

raise the temperature of the neon?


The figure reminds you of the

“bedspring model” of a solid

with particle-like atoms connected

by spring-like molecular bonds.

There are 3 degrees of freedom

associated with kinetic energy

+ 3 more associated with the

potential energy in the molecular

bonds = 6 degrees of freedom

total.

The energy stored in each degree of freedom is ½ NkBT,

so

Thermal Energy of a Solid

Compare theory with experiment

• Experimental values of CV for lead, aluminum, silicon, and

diamond are given in the figure.

• At high temperatures, CV

for each solid approaches

about 3R..

• At low temperatures, CV is

much less than 3R.



A gas consists of a vast number of molecules, each moving

randomly and undergoing millions of collisions every second.

Shown is the distribution of molecular speeds in a sample of

nitrogen gas at 20ºC.

The micro/macro

connection is built on

the idea that the

macroscopic properties

of a system, such as

temperature or pressure,

are related to the average

behavior of the atoms

and molecules.

Molecular Speeds and Collisions

Molecular speeds

• The Maxwell-Boltzmann distribution f (v) gives the

distribution of molecular speeds.


Molecular speeds

• The most probable speed for a given temperature is at the

peak of the curve.


Molecular speeds

• The function f (v) describing the actual distribution of

molecular speeds is called the Maxwell–Boltzmann

distribution.

• It can be derived from statistical mechanics considerations,

but that derivation is beyond our scope.

• Here is the result:


© 2016 Pearson Education, Inc.

A quantity of gas containing N molecules has a speed distribution

function f(v). How many molecules of this gas have speeds

between v1 and v2 > v1?

QuickCheck

A.

B.

C.

D.


Example 4

Find the most probable speed of the Maxwell-Boltzman

distribution function.


Example 5

For carbon dioxide gas (M = 44.0 g/mol) at 300 K calculate

the most probable speed, the average speed and the rms

speed.



For water vapor (M = 10.0 g/mol) at 400 K calculate the most

probable speed, the average speed and the rms speed.

Collisions between molecules · randomly and undergoing millions of collisions every second. Shown is the distribution of molecular speeds in a sample of nitrogen gas at 20ºC. The

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