Virial Equation Coefficients

Virial Equation Coefficients Volume explicit form: Z = 1 + B' P + C ' P2 + D ' P3 + " B C D Pressure explicit form: Z = 1 + + 2 + 3 + " V V V PV is defined as the compressibility factor where Z ≡ RT To relate B with B', C with C', … etc, P in volume explicit form is replaced by P = 2

ZRT . V

3

ZRT ⎛ ZRT ⎞ ⎛ ZRT ⎞ Z = 1 + B' P + C ' P + D ' P + " = 1 + B' + C' ⎜ ⎟ + D' ⎜ ⎟ +" V ⎝ V ⎠ ⎝ V ⎠ B' RT =1+ 1 + B' P + C ' P2 + D ' P3 + " V 2 2 C ' (RT ) + 1 + B' P + C ' P 2 + D ' P3 + " 2 V 3 3 D ' (RT ) + 1 + B' P + C ' P2 + D ' P 3 + " + " 3 V 2 3 ⎤ B' RT ⎡ ZRT ⎛ ZRT ⎞ ⎛ ZRT ⎞ =1+ + C' ⎜ ⎟ + D' ⎜ ⎟ + "⎥ ⎢1 + B' V ⎢⎣ V ⎝ V ⎠ ⎝ V ⎠ ⎥⎦ 2

C ' (RT ) + V2

2

3

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2 3 ⎡ ⎤ ZRT ⎛ ZRT ⎞ ⎛ ZRT ⎞ ' ' ' B C D + + + 1 ⎜ ⎟ ⎜ ⎟ + "⎥ ⎢ V ⎝ V ⎠ ⎝ V ⎠ ⎢⎣ ⎥⎦

2

2 3 3 ⎤ D ' (RT ) ⎡ ZRT ⎛ ZRT ⎞ ⎛ ZRT ⎞ " ' ' ' B C D + + + + + 1 ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ +" V 3 ⎣⎢ V ⎝ V ⎠ ⎝ V ⎠ ⎦⎥ 3

B' RT V 2 C ' (RT ) B' RT B' RT + + V2 V V

=1+

D ' (RT ) B' RT B' RT B' RT B ' RT ⎛ RT ⎞ C ' (RT ) RT C'⎜ B' + + + +" ⎟ +2 3 V V V V V V2 V ⎝ V ⎠ B' RT =1+ V 2 2 C ' (RT ) B'2 (RT ) + + V2 V2 3 3 3 D ' (RT ) B' C ' (RT ) C ' B' (RT ) + + +2 +" V3 V3 V3 ⎧ B = B' RT ⎪ 2 2 2 ⇒ ⎨C = C ' (RT ) + (B') (RT ) ⎪D = D ' (RT )3 + (B')3 (RT )3 + B' C ' (RT )3 + 2C ' B' (RT )3 ⎩ 3

2

2

⎧ B ⎪ B' = RT ⎪ 2 2 C − (B') (RT ) C − B2 ⎪ ⇒ ⎨C ' = = (RT )2 (RT )2 ⎪ 3 3 3 3 ⎪ D − (B') (RT ) − B' C ' (RT ) − 2C ' B' (RT ) D − B3 − B C − B2 − 2 C − B2 B D − 3BC + 2B3 ' D = = = ⎪ (RT )3 (RT )3 (RT )3 ⎩

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