Question

Derive the values of critical constants in terms of van der Waals constants.

Answer 1

Derivation of critical constants from the Van der Waals constants:
Van der Waals equation is,

`("P" + "an"^2/"V"^2) ("V" - "nb")` = nRT for 1 mole

From this equation, the values of critical constant P_CV_C and T_C arc derived in terms of a and b the Van der Waals constants.

`("P" + "an"^2/"V"^2) ("V" - "b")` = RT ..........(1)

On expanding the equestion (1)

`"PV" + "a"/"V" - "Pb" - "ab"/"V"^2 - "RT"` = 0 .........(2)

Multiplying eqestion (2) by `"V"^2/"P"`,

`"V"^2/"P"("PV" + "a"/"V" - "Pb" - "ab"/"V"^2 - "RT")` = 0

`"V"^3 + "aV"/"P" - "bV"^2 - "ab"/"P" - "RRV"^2/"P"` = 0 ........(3)

equation (3) is rearranged in the powers of V

`"V"^3 - ["RT"/"P" + "b"] "V"^2 + "aV"/"P" - "ab"/"P"` = 0 .........(4)

The above equation (4) is an cubic equation of V, which can have three roots. At the critical point. all the three values of V are equal to the critical volume V_C.

i.e. V = V_C.

V – V_C = 0 ……….(5)

(V – V_C)³ = 0 ………(6)
(V³ – 3V_CV² + 3V_C³V – V_C³ = 0 ………(7)

As equation (4) is identical with equation (7), comparing the ‘V’ terms in (4) and (7),

– 3V_CV² = `-["RT"_"C"/"P"_"C" + "b"]"V"^2` .......(8)

3V_C = `"b" + "RT"_"C"/"P"_"C"` ........(9)

3V_C² = `"a"/"P"_"C"` ........(10)

V_C³ = `"ab"/"P"_"C"` .....(11)

Divide equation (11) by (10)

`"V"_"C"^3/(3"V"_"C"^2) = ("ab"/"P"_"C")/("a"/"P"_"C")`

`"V"_"C"/3` = b

∴ V_C = 3b ......(12)

When equation (12) is substituted in (10)

3V_C² = `"a"/"P"_"C"`

P_C = `"a"/"3V"_"C"^2 = "a"/(3(3"b")^2) = "a"/(3 xx 9"b"^2) = "a"/(27"b"^2)`

∴ P_C = `"a"/(27"b"^2)` ......(13)

substituting the values of V_C and P_C in equation (9)

3V_C = `"b" + "RT"_"C"/"P"_"C"`

`3 xx 3"b" = "b" + "RT"_"C"/("a"/(27"b"^2))`

`9"b" - "b" ="RT"_"C"/"a" xx 27"b"^2`

8b = `("T"_"C"."R"27"b"^2)/"a"`

∴ T_C = `(8"ab")/(27"Rb"^2) = (8"a")/(27"RB")`

T_C = `(8"a")/(27"RB")` ....... (14)

Critical constant a and b can be calculated using Van der Waals Constant as follows:

a = 3V_C²P_C

b = `"V"_"C"/3` .......(15)