Heat capacity and Specific Heat

pp p

VV V

Q HC

dT T

Q UC

dT T

1.i

i

C Qc

n n dT

fixed P

fixed V

heat capacity per mole (or per gm…)i can correspond to either P or V.

Heat capacity and Specific Heat

f

i

pp p

p

T

f i pT

Q HC

dT T

dh c dT

h h h c dT

f

i

VV V

V

T

f i VT

Q UC

dT T

du c dT

u u u c dT

Usually the for real substances specific heats are functions of temperatureand we need to know ci (T) before we can evaluate h or u.

Heat capacity and Specific Heat

Q nc T

T

Q

small heat capacity

large heat capacity

What physical processes control the heat capacity of a substance?

Ideal Gas:

Monatomic gas; 3 translational DOF. Each contributes kBT/2 to the internal

energy. 3

2 A Bu N k T

Heat capacity and Specific Heat

3 3

2 2V A BV

duc N k R

dT

Monatomic Ideal Gas cont:

Diatomic Ideal Gas:

3 translational and two rotational DOF. Each contributes kBT/2 to the internalenergy.

5

2Vc R

Note that the specific heat of an ideal gas does not change with temperature.

Heat capacity and Specific Heat

Crystalline solids

What physical processes control the heat capacity of a substance?

Lattice vibrations ~ T 3

energy absorption by electrons ~ T

3

v v vD

Tc c lattice c electronic K T

T

3 translational DOF + 3 vibrational DOF 3R Law of Dulong and Petit.

Turns out that there is a temperature dependence to the specific heat .

K = 1940 J/mol-K and is material dependent

Low Temperatures

Aluminium 428 K

Cadmium 209 K

Chromium 630 K

Copper 343 K

Gold 165 K

Iron 470 K

Lead 105 K

TD

Heat capacity and Specific Heat

Also note that at very low T

v pc c

Law of Dulong and Petit

Heat capacity and Specific Heat

High temperatures

Specific heats at temperatures (~above room temperature) are usually expressedby an empirical equation of the form:

2 /pc a bT cT J mol K

Here, a, b and c are material dependent constants. Note that since the equationis empirical the constants have no simple physical interpretation and that suchequations should only be used over the temperature range for which the constantswere experimentally determined.

Heat capacity and Specific HeatRelation between cp and cV for an ideal gas

Vq du Pdv c dT Pdv

Pv RT

Pdv vdP RdT

V vq du Pdv c dT RdT vdP c R dT vdP

p vp p

q dhc c R dT

dT dT

p vc c R

Statement of the 1st Law

Ideal gas law (n = 1)

Total derivative

and .p Vp p V V

Q Q UC C

dT T dT

H

T

Recall the definitions of heat capacity

In terms of the specific internal energy u and enthalpy h the heat capacities per unit mass are called specific heats and take on the following forms:

and .p Vp p V V

q h q uc c

dT T dT T

The specific heat ratio, k, is a quantity that later we will find useful and isdefined by,

p

v

ck

c

Heat capacity and Specific Heat

Approximations for Condensed Phases

It turns out that the specific volume and specific internal energy of condensedphases vary little with pressure,

,

,

f

f

v T p v T

u T p u T

In the case of liquids the specific volume and internal energy may be evaluatedfor engineering purposes at the condensed liquid state at the temperature ofinterest. This is the incompressible substance model. It’san approximation.

Heat capacity and Specific Heat

A similar approximation can be made for the specific enthalpy. Since, h = u + pv,

, , ,

,

This equation for the enthalpy can be rewritten as,

,

,

f

f f

u T

f sat f f

f f sat

h T p u T p pv T p

h T p u T pv T

h T p h T p T v T pv T

h T p h T v T p p T

Heat capacity and Specific Heat

In the incompressible substance approximation since the specific internal energyand volume are taken as functions of temperature only,

( )v

duc T incompressible

dT

Note that this is an ordinary derivative since u is only a function of T in thisapproximation.

Also in this approximation, since taking derivatives whileholding p fixed yields,

( , ) ( )h T p u T pv

( ).p

h duincompressible

T dT

This says that in this model,

( ).p vc c incompressible

Heat capacity and Specific Heat

Then the changes in specific internal energy and enthalpy can be calculatedaccording to:

2

1

2

1

12 2 1

12 2 1 2 1 2 1 2 1

( )

( ) ( ) ( )

T

T

T

T

u u u c T dT

h h h u u v p p c T dT v p p

incompressible substance

Heat capacity and Specific Heat

The Ideal Gas Model

The equation of state of an ideal gas is given by where P is thepressure, V the volume, R the gas constant, n the number of moles and T the absolute temperature (K).

PV nRT

The gas constant R = 8.314 J/mole-K. There are various forms of this equationthat you should become familiar with.

Some text books are a little confusing on this issue. For example they define a symbol R different from the customary definition where the value of R is a constant for a particular gas of molecular weight M. Then the units of this R is kJ/kg-K. This is ugly since here R varies from material to materialsince the molecular weight is different for different materials. That is, R = R/M. This is ugly! It’s best just to know how to do the conversions.For example,

n

PV nRT

mass mPV RT RT

molecular weight M

V RTP

m MRT

pvM

The Ideal Gas Model

Also note that since R = 8.314 J/mole-K and 1 mole of a substance contains6.02 x 1023 molecules (Avogadro’s number, NA), R/NA= 1.38 x 10-23J/molecule ºK. = kB.kB is known as Boltzmann’s constant.

So, we can rewrite the ideal gas law as

A BA

RpV nRT nN T Nk T

N N is the no. of molecules

The Ideal Gas Model

Now, there’s a real interesting question to ask. How does the internal energy,or enthalpy of an ideal gas depend on the state of the gas, i.e., the P, V, T values?

Do you remember how an ideal gas is defined?

The molecules do not interact with one another. The ideal gas equation connectsthe 3 variables, so it’s not possible to hold two of them constant and alter the 3rd toexamine this problem.

It turns out that the internal energy of an ideal gas only depends on temperature.

( )

( ) ( )

p v

u u T

pv RT

h h T u T RT

c c R

The First Law and the Ideal Gas Model

du = q - w = q - pdv

For the ideal gas u = u(T) so, V

dudu dT c dT

dT and,

vq c dT pdv

Also for 1 mole of an ideal gas, pv = RT, pdv vdp RdT and substitution

(a)

for pdv in Equation (a),

pq c dT vdp (b)

In a quasi-static adiabatic process since, q = 0,

vpdV c dTpvdp c dT

The First Law and the Ideal Gas Model

Dividing the first equation by the second and rearranging,

; p p

v v

c cdp dv dvwhere

p c v v c

Integration yields, ln ln constant

or

constant

p v

pv