Thermodynamics

Thermodynamic Systems, States and Processes

Objectives are to:• define thermodynamics systems and states of systems• explain how processes affect such systems• apply the above thermodynamic terms and ideas to the laws of

thermodynamics

“Classical” means Equipartition Principle applies: each molecule has average energy ½ kT per in thermal equilibrium.

Internal Energy of a Classical ideal gas

At room temperature, for most gases:

monatomic gas (He, Ne, Ar, …) 3 translational modes (x, y, z)

kTEK2

3

diatomic molecules (N2, O2, CO, …) 3 translational modes (x, y, z) + 2 rotational modes (wx, wy)

kTEK2

5

pVkTNU2

3

2

3

Internal Energy of a Gas

pVkTNU2

3

2

3

A pressurized gas bottle (V = 0.05 m3), contains helium gas (an ideal monatomic gas) at a pressure p = 1×107 Pa and temperature T = 300 K. What is the internal thermal energy of this gas?

J105.705.0105.1 537 mPa

Changing the Internal Energy

U is a “state” function --- depends uniquely on the state of the system in terms of p, V, T etc.

(e.g. For a classical ideal gas, U = NkT )

WORK done by the system on the environment

Thermal reservoir

HEAT is the transfer of thermal energy into the system from the surroundings

There are two ways to change the internal energy of a system:

Work and Heat are process energies, not state functions.

Wby = -Won

Q

Work Done by An Expanding Gas

The expands slowly enough tomaintain thermodynamic equilibrium.

PAdyFdydW

Increase in volume, dV

PdVdW +dV Positive Work (Work isdone by the gas)

-dV Negative Work (Work isdone on the gas)

A Historical Convention

Energy leaves the systemand goes to the environment.

Energy enters the systemfrom the environment.

+dV Positive Work (Work isdone by the gas)

-dV Negative Work (Work isdone on the gas)

Total Work Done

PdVdW

f

i

V

V

PdVW

To evaluate the integral, we must knowhow the pressure depends (functionally)on the volume.

Pressure as a Function of Volume

f

i

V

V

PdVW

Work is the area underthe curve of a PV-diagram.

Work depends on the pathtaken in “PV space.”

The precise path serves to describe the kind of process that took place.

Different Thermodynamic Paths

The work done depends on the initial and finalstates and the path taken between these states.

Work done by a Gas

Note that the amount of work needed to take the system from one state to another is not unique! It depends on the path taken.

We generally assume quasi-static processes (slow enough that p and T are well defined at all times):

This is just the area under the p-V curve

f

i

V

Vby dVpW

V

p p

V

p

V

dWby = F dx = pA dx = p (A dx)= p dV

Consider a piston with cross-sectional area A filled with gas. For a small displacement dx, the work done by the gas is:

dx

When a gas expands, it does work on its environment

An Extraordinary Fact

The work done depends on the initial and finalstates and the path taken between these states.

BUT, the quantity Q - W does not dependon the path taken; it depends only on the initial and final states.

Only Q - W has this property. Q, W, Q + W,Q - 2W, etc. do not.

So we give Q - W a name: the internal energy.

-- Heat and work are forms of energy transfer and energy is conserved.

The First Law of Thermodynamics

(FLT)

U = Q + Won

work doneon the system

change intotal internal energy

heat added

to system

or

U = Q - Wby

State Function Process Functions

1st Law of Thermodynamics

• statement of energy conservation for a thermodynamic system

• internal energy U is a state variable• W, Q process dependent

system done work : positive

system addedheat : positive

by

to

W

Q

WQU

The First Law of Thermodynamics

bydWdQdE int

What this means: The internal energy of a systemtends to increase if energy is added via heat (Q)and decrease via work (W) done by the system.

ondWdQdE int

. . . and increase via work (W) done on the system.

onby dWdW

Efficiency of the engine (e) ratio of the work to the heat absorbed

= = 1-

The Carnot Cycle

Carnot Engine

• Carnot cycle run in forward direction

along abcda- Carnot Engine

ab

cd Q2

wQ1

Carnot Refrigerator

• Carnot cycle run in reverse direction

along adcba

ab

cd Q2

wQ2+w

In a reversible process the system changes in such a way that the system and surroundings can be put back in their original states by exactly reversing the process.

Changes are infinitesimally small in a reversible process.

= 1- = 1-

Efficiency of heat engine (e) in terms of absolute temperature

Carnot’s TheoremThe efficiency of any heat engine operating between two heat reservoirs of high and low temperatures is never greater than the efficiency of a Carnot engine; the efficiency of any reversible engine equals the efficiency of a Carnot engine.

• Irreversible processes cannot be undone by exactly reversing the change to the system.

• All Spontaneous processes are irreversible.• All Real processes are irreversible.

Entropy

• Entropy (S) is a term coined by Rudolph Clausius in the 19th century.

• Clausius was convinced of the significance of the ratio of heat delivered and the temperature at which it is delivered, Q

T

• Entropy can be thought of as a measure of the randomness of a system.

• It is related to the various modes of motion in molecules.

Second Law of Thermodynamics

The second law of thermodynamics: The entropy of the universe does not change for reversible processes

and

increases for spontaneous processes.

Reversible (ideal):

Irreversible (real, spontaneous):

Third Law of Thermodynamics

The third law of thermodynamics:

the entropy of a system at absolute zero is zero

.