Int-Cap-VK

7/29/2019 Int-Cap-VK

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Thermodynamics

Session 7

Determination of changes in internal energy and enthalpy:

No matter what is the process the internal energy and enthalpy are determined using the

following formulae.

dTCdH

dTCdU

P

V

If the process is isochoric, then .dQdTCdU V

If the process is isobaric, then .dQdTCdH P

Let us recall that1. For ideal gases internal energy is a function of temperature only.

2. The change in internal energy is a state function.

Let us take an adiabatic process and determine change in internal energy.

p

V

1

2


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Consider two states (1) and (2) on an adiabatic path. There is one isotherm for each state

as shown. Adiabatic curve at (1) is steeper than the isotherm. Whether the process

follows adiabatic path or another curve shown in the figure or any arbitrary path, as long

the initial and final state are same, we get the same change in internal energy.

Let us replace adiabatic path by two steps

1. 1 a constant volume process followed by

2. a

2 isothermal processto accomplish the same change in the state.

2121 aa UUU

a

p

V

1

2


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Since step 1 a is a constant volume process, dTCdU V will have to be used. The step

a 2 is isothermal and therefore dU = 0.

0

1

21

aT

T

VdTCU

Since ,TTa 2

2

1

21

T

T

VdTCU

Any process between any two states can be replaced by either an isothermal process

followed by constant volume process or constant volume process followed by an

isothermal process. The total change in internal energy is the change in internal energy

for constant volume step only. Thus no matter what process the change in internal energy

is given by dTCdU V .

Same idea may be extended to change in enthalpy.

Deviations of real gases form ideal behavior:

The deviation from ideal behavior is measured using what is known as

compressibility factor defined as the ratio of volume determined using ideal gas law

to the actual volume at any given temperature and pressure.

For ideal gases, Z = 1. Higher the value Z away from one higher will be the non-ideality.

Following figure gives the value of Z as a function pressure for different gases. All gasesapproach ideal gas behavior at low pressures. This figure is called compressibility chart.

There will be as many curves as there are number of gases. The chart that is given here is

for one temperature. Therefore this figure becomes highly cumbersome if applied for all

gases and all temperatures.

RT

pV

V

VZ

ideal

actual


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Theory of corresponding states:

It states that all fluids, when compared at the same reduced temperature and reducedpressure, will have same compressibility factor and all deviate from ideal behavior to the

same extent.

The reduced pressure and temperature are defined by

CH4N

2

Z

1.0

p

C2H6

C

r

C

r

TTT

p

pp


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Here all gases with the same reduced temperature and pressure lie on the same curve.

This chart is called generalized compressibility chart. Different gases are indicated by

different points on the curve. There will be as many curves as there are temperatures.

This is less cumbersome compared to compressibility chart.

The above table gives two gases with the same Z for different T and p. The reason is that

the critical values are different.

Z

pr

01.Tr

51.Tr 52.Tr

T p Tc p T pr

N2 189. 83.7 126. 33. 1. 2.

CH4 286.0 114. 190. 45. 1. 2.

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