Adiabatic Expansion (reversible and irreversible) Notes | EduRev

Chemistry Class 11

Class 11 : Adiabatic Expansion (reversible and irreversible) Notes | EduRev

The document Adiabatic Expansion (reversible and irreversible) Notes | EduRev is a part of the Class 11 Course Chemistry Class 11.
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ADIABATIC EXPANSION
In adiabatic expansion, no heat is allowed to enter or leave the system, hence, q = 0. When this value is substituted in first law of thermodynamics, ∆U= q + w, we get ∆U = w.
In expansion,

  • Work is done by the system on the surroundings, hence, w is negative.
  • Accordingly ∆U is also negative, i.e., internal energy decreases and therefore, the temperature of the system falls.

In case of compression,

  • ∆U is positive, i.e., internal energy increases and therefore,
  • The temperature of the system rises.

The molar specific heat capacity at constant volume of an ideal gas is given by.

Adiabatic Expansion (reversible and irreversible) Notes | EduRev

dU=Cv ·dT 
∆U=Cv∆T
So, w=∆U=Cv∆T
The value of ∆T depends upon the process whether it is reversible or irreversible.

REVERSIBLE ADIABATIC EXPANSION
Let P be the external pressure and ∆V the increase in volume. Thus, the work done by the system is
w = ∆U = – PdV

Adiabatic Expansion (reversible and irreversible) Notes | EduRev

We know CP – CV = R
Adiabatic Expansion (reversible and irreversible) Notes | EduRev
Adiabatic Expansion (reversible and irreversible) Notes | EduRev
Adiabatic Expansion (reversible and irreversible) Notes | EduRev
Adiabatic Expansion (reversible and irreversible) Notes | EduRev
Adiabatic Expansion (reversible and irreversible) Notes | EduRev
Adiabatic Expansion (reversible and irreversible) Notes | EduRev
Adiabatic Expansion (reversible and irreversible) Notes | EduRev

Adiabatic Expansion (reversible and irreversible) Notes | EduRev

Thus, knowing Ɣ, V1 , V2 and initial temperature, T1 , the final temperature, T2, can be readily evaluated.
Adiabatic Expansion (reversible and irreversible) Notes | EduRev
Thus, knowing Ɣ, P1, P2 and initial temperature, T1, the final temperature, T2, can be readily evaluated.

Adiabatic Expansion (reversible and irreversible) Notes | EduRev


IRREVERSIBLE ADIABATIC EXPANSION
In free expansion, the external pressure is zero, i.e. , work done is zero. Accordingly, ∆U which is equal to w is also zero. If ∆U  is zero, ∆T should be zero. Thus, in free expansion (adiabatically),
∆T = 0, ∆U= 0, w = 0 and ∆H = O.
In intermediate expansion, the volume changes from V1 to V2 against external pressure, Pext

Adiabatic Expansion (reversible and irreversible) Notes | EduRev
or
Adiabatic Expansion (reversible and irreversible) Notes | EduRev

Example 1. Two moles of an ideal monoatomic gas at NTP are compressed adiabatically and reversibly to occupy a volume of 4.48 dm3. Calculate the amount of work done, ∆U, final temperature and pressure of the gas. Cv for ideal gas 12.45J K -1 mol-1.
Solution. For an ideal gas,
Adiabatic Expansion (reversible and irreversible) Notes | EduRev
Initial volume, V= 2 × 22.4 = 44.8 dm3 
Initial pressure, P1 = 1 atm
Initial temperature, T1 = 273 K
Final volume, V2 = 4.48 dm3
Let the final pressure be Pand temperature be T
Applying

Adiabatic Expansion (reversible and irreversible) Notes | EduRev
or

Adiabatic Expansion (reversible and irreversible) Notes | EduRev

or

Adiabatic Expansion (reversible and irreversible) Notes | EduRev
P2 = (10)1.667(P1 = 1 given)
log P2 = 1.667 log 10= 1.667
P2 = antilog 1.667= 46.45 atm
Final temperature 

Adiabatic Expansion (reversible and irreversible) Notes | EduRev
= 1268 K

Work done on the system = n.Cv.ΔT
= 2 × 12.45 × (1268 - 273)
= 2 × 12.45 × 995 = 24775.5 J
From the first law of thermodynamics,
ΔE = q + w = 0 + 24775.5 = 24775.5 J

Example 2. A certain volume of dry air at NTP is expanded reversibiy to four times its volume (a) isothermally (b) adiabatically. Calculate thefinal pressure and temperature in each case, assuming ideal behaviour.
(CP \ CV for air = 1.4)
Solution. 
Let V1 be the initial volume of dry air at NTP.
(a) Isothermal expansion: During isothermal expansion, the temperature remains the same throughout. Hence, final temperature will be 273 K.
Since P1V1 = P2V2
Adiabatic Expansion (reversible and irreversible) Notes | EduRev
(b) Adiabatic expansion:
Adiabatic Expansion (reversible and irreversible) Notes | EduRev
Adiabatic Expansion (reversible and irreversible) Notes | EduRev
Adiabatic Expansion (reversible and irreversible) Notes | EduRev

CALCULATION OF ∆H, ∆U. WORK, HEAT ETC.
Case - 1 For an ideal gas undergoing a process. the formula to be used are:

Adiabatic Expansion (reversible and irreversible) Notes | EduRev

dH = dU + d(PV)
ΔH = ΔU + nRΔT
du = dQ + dw

Calculation of q, w , ∆H, ∆U for a reversible isothermal process involving an ideal gas :
ΔU = q + w = 0 ⇒ -w

Adiabatic Expansion (reversible and irreversible) Notes | EduRev

Calculation of q, w, ∆H, ∆U for an Irreversible isothermal process involving an ideal gas:
For isothermal process involving
ΔH = ΔU = 0 ∵ ΔT = 0
Also,

Adiabatic Expansion (reversible and irreversible) Notes | EduRev

Adiabatic Expansion (reversible and irreversible) Notes | EduRev 
For isobaric process

Adiabatic Expansion (reversible and irreversible) Notes | EduRev
Adiabatic Expansion (reversible and irreversible) Notes | EduRev

Adiabatic Expansion (reversible and irreversible) Notes | EduRev

= -nR(T2 - T1) (∵ Pext = P)
= -nRΔT

Adiabatic Expansion (reversible and irreversible) Notes | EduRevCalculation of q, w, ∆H, ∆U for an IRREVERSIBLE ISOCHORIC process involving an ideal gas:

Adiabatic Expansion (reversible and irreversible) Notes | EduRev
w= 0   ∵ dV = 0

Adiabatic Expansion (reversible and irreversible) Notes | EduRevCalculation of q, w, ∆H, ∆U for reversible adiabatic processAdiabatic Expansion (reversible and irreversible) Notes | EduRevFor an adiabatic process,
dq = 0 ⇒ dU = dw

Adiabatic Expansion (reversible and irreversible) Notes | EduRev

for a reversible change

Adiabatic Expansion (reversible and irreversible) Notes | EduRev
Adiabatic Expansion (reversible and irreversible) Notes | EduRev

Adiabatic Expansion (reversible and irreversible) Notes | EduRev
Adiabatic Expansion (reversible and irreversible) Notes | EduRev
Adiabatic Expansion (reversible and irreversible) Notes | EduRev
Now substituting V = nRT/P in equation

Adiabatic Expansion (reversible and irreversible) Notes | EduRev

substituting T = PV/nR in eq...

Adiabatic Expansion (reversible and irreversible) Notes | EduRev
⇒ Equation (i), (ii) and (iii) is valid only for reversible adiabatic process, for irreversible adiabatic process these equations are not applicable.

  • Expression for work:

Adiabatic Expansion (reversible and irreversible) Notes | EduRev

  • Expression for ΔH and ΔU

Adiabatic Expansion (reversible and irreversible) Notes | EduRev


Adiabatic Expansion (reversible and irreversible) Notes | EduRev


CALCULATION OF Q, W, ∆H, ∆U FOR IRREVERSIBLE ADIABATIC 
PROCESS INVOLVING AN IDEAL GAS:
Operation wise adiabatic process and isothermal process are similar hence all the criteria that is used for judging an isothermal irreversible process are applicable to adiabatic process.

If large amount of dust particles are removed abruptly an irreversible adiabatic expansion take place.

In an irreversible adiabatic process, an ideal gas is subjected to compression or expansion in a thermally insulated vessel.

The heat absorbed in the process =0
⇒ dU = wirr ....(i)
Adiabatic Expansion (reversible and irreversible) Notes | EduRev
If Pext, = P2 = Pfinal
Then

Adiabatic Expansion (reversible and irreversible) Notes | EduRev
eq. (ii) or (iii) can be solved for T2
Expression for w

Adiabatic Expansion (reversible and irreversible) Notes | EduRev

Note: If two states A and B are connected by a reversible path then they can never be connected by an irreversible path.
Adiabatic Expansion (reversible and irreversible) Notes | EduRevIf the two states are linked by an adiabatic reversible and irreversible path then wrev = ∆Urev
But as U is a state function
Therefore, ∆U irrev = ∆Urev
wirrev = wrev
as work is a path function.

If we assume that
wirrev = wrev

It implies that   which again is a contradiction as U is a state function.
∆U irrev ≠ ∆Urev
Two states A and B can never lie both on a reversible as well as irreversible adiabatic path.

There lies only one unique adiabatic path linkage between two states A and B.

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