State Postulate

According to the State Postulate the number of intensive variable needed to specify all other intensive variables equals the number of relevant, reversible work modes plus one.

Here there is one such mode (mechanical compression); i.e., two intensive variables define the system.

Equations of State If there are other work modes (electrical

or magnetic forces) we need one additional variable per mode.

It requires one additional variable, which must be extensive to determine all other extensive variables.

Use equations of state to determine unknown variables from those known.

Thermally Perfect Gases

A thermally perfect gas is one whose molecules have negligible volume and do not interact with each other.

pV = nRT

p = cRT

pV = mRT

p = RT

Compressibility Compressibility Z is defined as:

Z pRTPerfect Gas Z = 1Imperfect Gas Z < 1Gases are not thermally perfect at higher pressures.

For a perfect gas it can also be shown that U = U(T only)

Properties of a Gas

In general let’s say u = u(T,V), then

duuT V) dT

uV T) dV

For a thermally perfect gas:

dudu

dT V) dT

Specific Heat The specific heat is the amount of heat

required to raise the substance by one degree.

Since heat is not a state variable, path must be specified.

For gases we use cv and cp. Not valid at phase change, with work.

Relationship between cv & cp

In general:

For perfect gas:

If cv & cp constant, gas calorically perfect

If thermally and calorically perfect, gas is ideal

ducvdT dT

dv)T dv

ducvdT

Enthalpy

The expression u + pV occurs frequently

Define the enthalpy as :

h = u + pV

Since for a perfect gas pV = RT

h = h(T only)

Properties of a Gas

In general let’s say h = h(T,p), then

dh hT p) dT

hp T) dp

For a thermally perfect gas:

dh dh

dT Vp) dT

Relationship between cv & cp

In general:

For perfect gas:

dh c pdT dT

dp)T dp

dhcpdT

dh du d(pV )cvdT d(RT)cvdT RdT c pdT

cp cv R

Ratio of Specific Heats

Definition:

For perfect gas:

Also:

1 < < 1.67, for air = 1.4

c pcv

cv R

1

c p

1R

Incompressible Substances Liquids or solids (e.g., water =0.1 %

for a change from 1 atm. to 50 atm.)

Generally

ducvdT dT

dv)T dv

If incompressible v-constant

then

ducvdT

Incompressible Substances

Now: dh = du + d(pv) = du + pdV =Vdp

dh = cvdT +Vdp

Compare with: h = h(t,p) then

Compare dT terms: cp = cv = c

Only one value of specific heat

dh hT|p dT

hp|T dpc pdT

hp|T dp