Atkins’ Physical ChemistryEighth Edition

Chapter 2 – Lecture 4The First Law

Copyright © 2006 by Peter Atkins and Julio de Paula

Peter Atkins • Julio de Paula

State Functions and Exact Differentials

• State function – state of the system independent of mannerand time to prepare

• Path functions – processes that describe prep of the state

• e.g., Heat, work, temperature

• Systems do not possess heat, work, or temperature

• Advantage of state functions:

• Can combine measurements of different properties toobtain value of desired property

Fig 2.20 As V and T change, the internal energy, U changes

adiabatic

non-adiabatic

f

i

dUUΔ

When the systemcompletes the path:

dU is said to be:an exact differential

f

path,i

dqq

q is said to be:an inexact differential

Fig 2.21 The partial derivative (∂U/∂V)T is the slope of Uw.r.t V at constant T

Fig 2.22 The partial derivative (∂U/∂T)V is the slope of Uw.r.t T at constant V

Fig 2.23 The overall change in U (i.e. dU) which arises whenboth V and T change

dTT

UdV

V

UdU

VT

The full differential of Uw.r.t V and T

Partial derivatives representing physical properties:

VV T

UC

PP T

HC

Constant volume heat capacity

Constant pressure heat capacity

Internal pressureT

T V

U

π

Fig 2.24 The internal pressure πT is the slope of U w.r.t. Vat constant T

has dimensions of pressure

Partial derivatives representing physical properties:

VV T

UC

PP T

HC

Constant volume heat capacity

Constant pressure heat capacity

Internal pressureT

T V

U

π

dTCdVdTT

UdV

V

UdU VT

VT

πNow:

Two cases for real gases:

0V

U

TT

π

Fig 2.25 For a perfect gas the internal energy is independentof volume at constant temperature

Fig 2.26 Schematic of Joule’s attempt to measure ΔU foran isothermal expansion of a gas

TT V

U

π

at 22 atm

q absorbed by gas ∝ T

• For expansion into vacuum, w = 0

• No ΔT was observed,so q = 0

• In fact, experiment was crudeand did not detect thesmall ΔT

Changes in Internal Energy at Constant Pressure

• Partial derivatives useful to obtain a property that cannotbe observed directly:

• e.g.,

• Dividing through by dT:

• Where expansion coefficient

• and isothermal compressibility

dTCdVdU VT π

VP

TP

CT

V

T

U

π

PT

V

V

1

α

TT P

V

V

1

κ

VP

TP

CT

V

T

U

π

Changes in Internal Energy at Constant Pressure

VTP

CVT

U

απ

PT

V

V

1

α

TT P

V

V

1

κ

Substituting:

Into:

Gives:

For a perfect gas: VP

CT

U

Changes in Internal Energy at Constant Pressure

• May also express the difference in heat capacities with observables:

• Cp – Cv = nR

• Since H = U + PV = U + nRT

• And:

PPVP T

U

T

HCC

nRT

HnR

T

UCC

PPVP

T

2

VPTV

CCκ

α