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Relationship between
vanderwaals equation and
critical state
CECH-202
UNIT-1(A)
BY - Ms MAYURI R SOMPURA
• For ideal gas
PV=nRT
• Vanderwaals equation :
(ideal gas equation with correction factor )
𝒑 +
𝒂
𝑽𝒎𝟐 𝐕𝐦 − 𝐛 = 𝐑𝐓
For n moles :-
𝒑 +
𝒏𝒂
𝑽𝒎𝟐 𝐕𝐦 − 𝐧𝐛 = 𝐑𝐓
BY - Ms MAYURI R SOMPURA
𝒑𝑽𝒎 − 𝒑𝒃 +
𝒂
𝑽𝒎
−
𝒂𝒃
𝑽𝒎 𝟐 − 𝐑𝐓 = 𝟎
• Multiply by 𝑽𝒎 𝟐
• Then divide by P
• Rearrange the equation in decreasing powers of Vm
(𝑽𝒎)𝟑
− 𝒃 +
𝑹𝑻
𝑷
𝑽𝒎 𝟐
+
𝒂
𝑷
𝑽𝒎 −
𝒂𝒃
𝑷
= 𝟎
BY - Ms MAYURI R SOMPURA
(𝑽𝒎)𝟑− 𝒃 +
𝑹𝑻
𝑷
𝑽𝒎 𝟐 +
𝒂
𝑷
𝑽𝒎 −
𝒂𝒃
𝑷
= 𝟎
• This is a cubic equation in variable Vm.
• Hence,for a single set of value of P and T there should be three vales
of Vm.
• Out of three, only one may be real and other two may be imaginery.
• Graph :-
At constant temperature, values of V are plotted against P
BY - Ms MAYURI R SOMPURA
BY - Ms MAYURI R SOMPURA
• It can be seen that for pressure there are 3 values of Vm marked.
• As the temperature increases, the isotherm moves up.
• The three values of V gets closer to one another.
• At certain temperature, there is only one real value for V
corresponding to pressure.
• Small portions of b-c and f-e represent super – saturated vapour and
super heated liquid respectively.
• As temperature is raised loops become smaller and smaller.
• At a critical point all three values become identical.
• Since temperature is critical, volume represents critical volume.
𝑽𝒎 = 𝑽𝒎, 𝒄
BY - Ms MAYURI R SOMPURA
𝑽𝒎 = 𝑽𝒎, 𝒄 𝐎𝐑 (𝑽𝒎 − 𝑽𝒎, 𝒄)𝟑= 𝟎
• Expand the equation and write in decreasing powers of Vm.
(𝑽𝒎)𝟑− 𝟑𝑽𝒎, 𝒄 𝑽𝒎 𝟐 + 𝟑 𝑽𝒎, 𝒄 𝟐𝑽𝒎 − 𝑽𝒎, 𝒄 𝟑 = 𝟎
• This equation must be identical to vanderwaals equation at critical
temperature and pressure for one mole of the gas.
(𝑽𝒎)𝟑− 𝒃 +
𝑹𝑻𝒄
𝑷𝒄
𝑽𝒎 𝟐 +
𝒂
𝑷𝒄
𝑽𝒎 −
𝒂𝒃
𝑷𝒄
= 𝟎
• Hence, coefficients of equal powers of Vm should be equal to one
another:
BY - Ms MAYURI R SOMPURA
𝑽𝒎 = 𝑽𝒎, 𝒄 𝐎𝐑 (𝑽𝒎 − 𝑽𝒎, 𝒄)𝟑= 𝟎
• Expand the equation and write in decreasing powers of Vm.
(𝑽𝒎)𝟑− 𝟑𝑽𝒎, 𝒄 𝑽𝒎 𝟐 + 𝟑 𝑽𝒎, 𝒄 𝟐𝑽𝒎 − 𝑽𝒎, 𝒄 𝟑 = 𝟎
• This equation must be identical to vanderwaals equation at critical
temperature and pressure for one mole of the gas.
(𝑽𝒎)𝟑− 𝒃 +
𝑹𝑻𝒄
𝑷𝒄
𝑽𝒎 𝟐 +
𝒂
𝑷𝒄
𝑽𝒎 −
𝒂𝒃
𝑷𝒄
= 𝟎
• Hence, coefficients of equal powers of Vm should be equal to one
another:
BY - Ms MAYURI R SOMPURA
• 𝟑𝑽𝒎, 𝒄 = 𝒃 +
𝑹𝑻𝒄
𝑷𝒄
• 𝟑 𝑽𝒎, 𝒄 𝟐
=
𝒂
𝑷𝒄
• 𝑽𝒎, 𝒄 𝟑 =
𝒂𝒃
𝑷𝒄
BY - Ms MAYURI R SOMPURA
Relation between critical constants and
Van der Waal’s constants
=
C
8a
T
27Rb
=
c 2
a
P
27b
Vm,c = 3b
Boyle temperature(TB) =
B
a
T
Rb
Critical temperature
Critical pressure
Critical volume
• Thus, knowing critical constants of gas, it is possible to calculate
vanderwaals constants and vice versa.
➢Assignment :-
➢Derive the relationship between vanderwaals equation and critical
constant.
BY - Ms MAYURI R SOMPURA

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Relationship between vanderwaals equation and critical state converted

  • 1. Relationship between vanderwaals equation and critical state CECH-202 UNIT-1(A) BY - Ms MAYURI R SOMPURA
  • 2. • For ideal gas PV=nRT • Vanderwaals equation : (ideal gas equation with correction factor ) 𝒑 + 𝒂 𝑽𝒎𝟐 𝐕𝐦 − 𝐛 = 𝐑𝐓 For n moles :- 𝒑 + 𝒏𝒂 𝑽𝒎𝟐 𝐕𝐦 − 𝐧𝐛 = 𝐑𝐓 BY - Ms MAYURI R SOMPURA
  • 3. 𝒑𝑽𝒎 − 𝒑𝒃 + 𝒂 𝑽𝒎 − 𝒂𝒃 𝑽𝒎 𝟐 − 𝐑𝐓 = 𝟎 • Multiply by 𝑽𝒎 𝟐 • Then divide by P • Rearrange the equation in decreasing powers of Vm (𝑽𝒎)𝟑 − 𝒃 + 𝑹𝑻 𝑷 𝑽𝒎 𝟐 + 𝒂 𝑷 𝑽𝒎 − 𝒂𝒃 𝑷 = 𝟎 BY - Ms MAYURI R SOMPURA
  • 4. (𝑽𝒎)𝟑− 𝒃 + 𝑹𝑻 𝑷 𝑽𝒎 𝟐 + 𝒂 𝑷 𝑽𝒎 − 𝒂𝒃 𝑷 = 𝟎 • This is a cubic equation in variable Vm. • Hence,for a single set of value of P and T there should be three vales of Vm. • Out of three, only one may be real and other two may be imaginery. • Graph :- At constant temperature, values of V are plotted against P BY - Ms MAYURI R SOMPURA
  • 5. BY - Ms MAYURI R SOMPURA
  • 6. • It can be seen that for pressure there are 3 values of Vm marked. • As the temperature increases, the isotherm moves up. • The three values of V gets closer to one another. • At certain temperature, there is only one real value for V corresponding to pressure. • Small portions of b-c and f-e represent super – saturated vapour and super heated liquid respectively. • As temperature is raised loops become smaller and smaller. • At a critical point all three values become identical. • Since temperature is critical, volume represents critical volume. 𝑽𝒎 = 𝑽𝒎, 𝒄 BY - Ms MAYURI R SOMPURA
  • 7. 𝑽𝒎 = 𝑽𝒎, 𝒄 𝐎𝐑 (𝑽𝒎 − 𝑽𝒎, 𝒄)𝟑= 𝟎 • Expand the equation and write in decreasing powers of Vm. (𝑽𝒎)𝟑− 𝟑𝑽𝒎, 𝒄 𝑽𝒎 𝟐 + 𝟑 𝑽𝒎, 𝒄 𝟐𝑽𝒎 − 𝑽𝒎, 𝒄 𝟑 = 𝟎 • This equation must be identical to vanderwaals equation at critical temperature and pressure for one mole of the gas. (𝑽𝒎)𝟑− 𝒃 + 𝑹𝑻𝒄 𝑷𝒄 𝑽𝒎 𝟐 + 𝒂 𝑷𝒄 𝑽𝒎 − 𝒂𝒃 𝑷𝒄 = 𝟎 • Hence, coefficients of equal powers of Vm should be equal to one another: BY - Ms MAYURI R SOMPURA
  • 8. 𝑽𝒎 = 𝑽𝒎, 𝒄 𝐎𝐑 (𝑽𝒎 − 𝑽𝒎, 𝒄)𝟑= 𝟎 • Expand the equation and write in decreasing powers of Vm. (𝑽𝒎)𝟑− 𝟑𝑽𝒎, 𝒄 𝑽𝒎 𝟐 + 𝟑 𝑽𝒎, 𝒄 𝟐𝑽𝒎 − 𝑽𝒎, 𝒄 𝟑 = 𝟎 • This equation must be identical to vanderwaals equation at critical temperature and pressure for one mole of the gas. (𝑽𝒎)𝟑− 𝒃 + 𝑹𝑻𝒄 𝑷𝒄 𝑽𝒎 𝟐 + 𝒂 𝑷𝒄 𝑽𝒎 − 𝒂𝒃 𝑷𝒄 = 𝟎 • Hence, coefficients of equal powers of Vm should be equal to one another: BY - Ms MAYURI R SOMPURA
  • 9. • 𝟑𝑽𝒎, 𝒄 = 𝒃 + 𝑹𝑻𝒄 𝑷𝒄 • 𝟑 𝑽𝒎, 𝒄 𝟐 = 𝒂 𝑷𝒄 • 𝑽𝒎, 𝒄 𝟑 = 𝒂𝒃 𝑷𝒄 BY - Ms MAYURI R SOMPURA
  • 10. Relation between critical constants and Van der Waal’s constants = C 8a T 27Rb = c 2 a P 27b Vm,c = 3b Boyle temperature(TB) = B a T Rb Critical temperature Critical pressure Critical volume
  • 11. • Thus, knowing critical constants of gas, it is possible to calculate vanderwaals constants and vice versa. ➢Assignment :- ➢Derive the relationship between vanderwaals equation and critical constant. BY - Ms MAYURI R SOMPURA