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phys documents (dragged) 28 - p cr V m, cr /RT cr = 3 8 ,...

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Chapter 7: Statistical physics 31 7.3 Pressure on a wall The number of molecules that collides with a wall with surface A within a time τ is given by: ±± d 3 N = ± 0 π ± 0 2 π ± 0 nAv τ cos( θ ) P ( v, θ , ϕ ) dvd θ d ϕ From this follows for the particle ±ux on the wall: Φ = 1 4 n ± v ² . For the pressure on the wall then follows: d 3 p = 2 mv cos( θ ) d 3 N A τ , so p = 2 3 n ± E ² 7.4 The equation of state If intermolecular forces and the volume of the molecules can be neglected then for gases from p = 2 3 n ± E ² and ± E ² = 3 2 kT can be derived: pV = n s RT = 1 3 Nm ² v 2 ³ Here, n s is the number of moles particles and N is the total number of particles within volume V . If the own volume and the intermolecular forces cannot be neglected the Van der Waals equation can be derived: ´ p + an 2 s V 2 µ ( V - bn s )= n s RT There is an isotherme with a horizontal point of in±ection. In the Van der Waals equation this corresponds with the critical temperature, pressure and volume of the gas. This is the upper limit of the area of coexistence between liquid and vapor. From dp/dV =0 and d 2 p/dV 2 =0 follows: T cr = 8 a 27 bR ,p cr = a 27 b 2 ,V cr =3 bn s For the critical point holds:
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Unformatted text preview: p cr V m, cr /RT cr = 3 8 , which differs from the value of 1 which follows from the general gas law. Scaled on the critical quantities, with p * := p/p cr , T * = T/T cr and V * m = V m /V m, cr with V m := V/n s holds: p * + 3 ( V * m ) 2 ( V * m-1 3 ) = 8 3 T * Gases behave the same for equal values of the reduced quantities: the law of the corresponding states . A virial expansion is used for even more accurate views: p ( T,V m ) = RT 1 V m + B ( T ) V 2 m + C ( T ) V 3 m + The Boyle temperature T B is the temperature for which the 2nd virial coefcient is 0. In a Van der Waals gas, this happens at T B = a/Rb . The inversion temperature T i = 2 T B . The equation of state for solids and liquids is given by: V V = 1 + p T- T p = 1 + 1 V V T p T + 1 V V p T p...
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