Ideal gas and kinetic gas theory

pV nRT

AR N k231.38 10J

kK

Boltzmann constant

pV NkT

Ideal gas and kinetic gas theory

pV NkTProperties of an ideal gas:

1. Large distances between particles

2. Particles interact only during collisions (“hard spheres”)

2

3pV NKE

3

2KE kT

Temperature and average translational kinetic energy

• Find the average kinetic energy for one nitrogen molecule at room temperature.

• To which speed does this correspond?• How much translational kinetic energy does this

molecule have at 0 K?• How much energy is stored in this auditorium in

translational kinetic energy of the air molecules?

3

2KE kT

Argon Helium

Same kinetic energy, hence 43.1

40He

Ar

v

v

What will the ratio of the velocities be after mixing ?

Maxwell velocity distribution

200 400 600 800 1000 1200

Occ

urre

nce

Speed (m/s)

Argon, 300 K

Vrms=432 m/s

Vmean=398 m/sVmax=353 m/s

Maxwell velocity distribution

200 400 600 800 1000 1200

Occ

urre

nce

Speed (m/s)

Argon, 300 K

Neon, 300 K

Helium, 300 K

Thermodynamic work and the first law of thermodynamics

Ideal gas

p, T, V, n

Internal energy UKinetic energy of particles

System

Environment

Ideal gas

System

Environment

System

Environment

dxF=pA

F=pA dW Fdx

dW pAdx

dW pdV

Thermodynamic work –definitions

• Work is done if the boundary of the system shifts with respect to the environment

• dV > 0: expansion “work by gas on env.”

• dV < 0: compression “work on gas by env.”

2

12

1

W pdVdW pdV

An isobaric processp

V

Heat Q

1

2

1 2

2

12

1

W pdV 2 1p V V

Did the temperature change as well?

P=1 atm

2L balloon

An isochoric processp

V

Heat Q

1

2

1

2

2

12

1

W pdV 0

Did the temperature change?

An isothermal processp

V

1

2

1

2

2

12

1

W pdV2

1

nRTdV

VIce water

Ice water

Is the work negative or positive?

2

1

lnV

W nRTV

10% reduction

Was there any heat exchange?

Insulation

Insulation

An adiabatic processp

V

1

2

1

2

2

12

1

W pdV ?

No heat transfer possible

What is a quasistatic process?

1

2

Ice water

Ice water

p

V

1

2

Temperature is always equal to the temperature of the bathThermal equilibrium at all times!= quasistatic= reversible

Realistic: it takes time for temperature equilibration –

not thermal equilibrium at all times

= not quasistatic= irreversible

Processes you should be able to distinguish:

• Isobaric• Isothermal• Isochoric• Adiabatic• Quasistatic• Reversible• Irreversible

Internal energy

• Thermal internal energy: part that changes

if T changes

• Can be chemical, nuclear, strain, PE, KE

• Ideal gas: total KE of all particles

• Symbol: U

• Property: characterizes “state” of system

Internal energy of ideal gas?

Monatomic gas:Each particle has 3 degrees of freedom

Each degree of freedom has ½ kT of (kinetic) energy on average

3

2U NkT

Diatomic gas:Each molecule has 3 degrees of freedom for translation

Each molecule has 2 degrees of freedom for rotation

Each molecule has 1 degree of freedom for bond stretching which engages only at very high temperatures

Each degree of freedom has ½ kT of (kinetic) energy on average

5

2U NkT

Example

• Compare the internal energy of 1 mole of

helium and 1 mole of nitrogen at 300 K

First Law of thermodynamics

System in state 1:p1, T1, V1, U1

System in state 2:p2, T2, V2, U2

process

Heat exchange Q

Work W

Conservation of energy:

2 1U U Q W

Isobaric processp

V

Heat Q

1

2

1 2

12 203W J

P=1 atm

2L balloon

How much heat transfer was necessary to the 7-L flask of nitrogen in order to inflate the 2-L balloon in an isobaric process?

An isothermal processp

V

1

2

1

2

Ice water

Ice water

2

1

lnV

W nRTV

10% reduction

How much ice has been molten in the ice water during this compression? The flask contains originally 7 L of nitrogen at initially 1 atm.

Adding heat -how much is needed ?

Q Q

T increases T increases

p increases p constant

V constant V increases

2 1Q U U W 2 1Q U U

2 12

fQ nR T T 2 1 2 12

fQ nR T T p V V