Gráficas del fluido de van der Waals

Este es  un archivo para el programa Mathematica por lo que se dejan indicadas las instrucciones.

La energía libre de Helmholtz (molar) de van der Waals es

f = - RT [ ln ((v-b)/((Subscript[N, 0])(λ^3)))+1] - (a/v)       con     λ=(h/(Sqrt[2πmkT]))
Siempre que se tenga que realizar un cálculo numérico, primero hay que adimensionalizar. Para hacerlo, se necesitan tres cantidades para formar con ellas las unidades de masa, tiempo y longitud. Esto es equivalente a dar valores arbitrarios a tres constantes. Las demás cantidades quedan entonces expresadas en términos de ellas. Escogemos entonces,

                                          R = 1, b =1, a = 10.

Como tenemos la libertad de escoger la masa del átomo en cuestión,usamos 

				(Subscript[N, 0])((((h/(Sqrt[2πmk]))))^3) = 1
 Por lo tanto,la energía libre de Helmholtz adimensionalizada es

f = - t [ ln (v - 1) + (3/2)ln (t) + 1] - (10/v)

con t y v la temperatura y volumen adimensionalizados.

La presión es

p = (t/(v - 1))-(10/(v^2))

El potencial químico es

μ = f + p v

El punto crítico es

(Subscript[p, c ])= (10/27)     (Subscript[v, c])= 3     (Subscript[T, c])=(80/27)

Definiciones

p[v_,t_] = t/(v - 1) - 10/v^2;

f[v_,t_] = - t (Log[v - 1] + 1.5*Log[t] + 1) - 10/v;

g[v_,t_,p_] = f[v,t] + p*v;

tc = 80/27;
pc = 10/27;

Figura 1

Isotermas de van der Waals para T> (Subscript[T, c])(AZUL), T= (Subscript[T, c]) (VERDE) y T< (Subscript[T, c])(ROJO)

Plot[{p[v,2.7],p[v,tc],p[v,3.2]},{v,1.5,10}, AxesOrigin -> {0.0,0.17},  TextStyle->{FontFamily->"Times",FontSize->14},AxesLabel-> TraditionalForm/@{v,p},PlotStyle-> {{RGBColor[1,0,0],Thickness[0.005]},{RGBColor[0,1,0],Thickness[0.005]},{RGBColor[0,0,1],Thickness[0.005]}}, AxesStyle-> Thickness[0.004]]

Figura 2

Isoterma de van der Waals para T< (Subscript[T, c])    (temperatura, t = 2.7)
Cálculo de ((Subscript[p, max]),(Subscript[v, max])) y ((Subscript[p, min]),(Subscript[v, min]))   (((∂p)/(∂v)))T = 0

Solve[D[p[v,2.7],v]==0,v]

vmin = 2.19426;
vmax =4.45547;

pmax=p[vmax,2.7];
pmin =p[vmin,2.7];

Plot[{p[v,2.7],pmax,pmin},{v,1.1,13}, AxesOrigin -> {0.,0.16}, TextStyle->{FontFamily->"Times",FontSize->14},AxesLabel-> TraditionalForm/@{v,p},PlotStyle->{ Thickness[0.005],Thickness[0.001],Thickness[0.001]}
, AxesStyle-> Thickness[0.004]]

Figura 3

Isoterma de van der Waals para T< (Subscript[T, c])    (temperatura, t = 2.7)
Cálculo de (Subscript[p, coex]), (Subscript[v, l]) y (Subscript[v, g]) .Tales  cantidades corresponde a la construcción de Maxwell de áreas iguales. Sin embargo, esto es equivalente a resolver simultánemante,
p((Subscript[v, l]),T) = p((Subscript[v, g]),T)  y  μ((Subscript[v, l]),T) = μ((Subscript[v, g]),T)

FindRoot[{f[x,2.7] + p[x,2.7]*x-f[y,2.7]-p[y,2.7]*y==0,p[x,2.7]-p[y,2.7]==0},{x,1.8},{y,8}]

vl = 1.85349;
vg = 6.59637;

checando ...

p[vg,2.7]

p[vl,2.7]

pcoex = p[vg,2.7];

Plot[{p[v,2.7],pcoex},{v,1.2,13}, AxesOrigin -> {0.,0.16}, TextStyle->{FontFamily->"Times",FontSize->14},AxesLabel-> TraditionalForm/@{v,p},PlotStyle->{ Thickness[0.005],Thickness[0.001]}
, AxesStyle-> Thickness[0.004]]

Figura 4

g(v;T,p) = f(v,T) + p v  para T> (Subscript[T, c])    (temperatura t = 3.2, presión p = 0.5)

Plot[g[v,3.2,0.5],{v,1.3,10},AxesOrigin -> {0.,-11.8}, TextStyle->{FontFamily->"Times",FontSize->14},AxesLabel-> TraditionalForm/@{v,g},PlotStyle->{ Thickness[0.005]}
, AxesStyle-> Thickness[0.004]]

Figura 5  (tres figuras)

g(v;T,p) = f(v,T) + p v  para T= (Subscript[T, c]) y p> (Subscript[p, c])  (temperatura tc = 80/27, presión p = 0.5)

Plot[g[v,tc,0.5],{v,1.2,10},AxesOrigin -> {0.,-10.3}, TextStyle->{FontFamily->"Times",FontSize->14},AxesLabel-> TraditionalForm/@{v,g},PlotStyle->{ Thickness[0.005]}
, AxesStyle-> Thickness[0.004]]

g(v;T,p) = f(v,T) + p v  para T= (Subscript[T, c])  y p= (Subscript[p, c])   (temperatura tc = 80/27, presión pc = 10/27)

Note que el mínimo ocurre en vc = 3

Plot[g[v,tc,pc],{v,1.2,15},AxesOrigin -> {0.,-11.5}, TextStyle->{FontFamily->"Times",FontSize->14},AxesLabel-> TraditionalForm/@{v,g},PlotStyle->{ Thickness[0.005]}
, AxesStyle-> Thickness[0.004]]

g(v;T,p) = f(v,T) + p v  para T= (Subscript[T, c]) y p< (Subscript[p, c])  (temperatura tc = 80/27, presión p = 0.3)

Plot[g[v,tc,0.3],{v,1.2,10},AxesOrigin -> {0.,-11.7}, TextStyle->{FontFamily->"Times",FontSize->14},AxesLabel-> TraditionalForm/@{v,g},PlotStyle->{ Thickness[0.005]}
, AxesStyle-> Thickness[0.004]]

Figura 6  (seis figuras)

g(v;T,p) = f(v,T) + p v  para T< (Subscript[T, c]) y p> (Subscript[p, max])  (temperatura t =2.7, presión p = 0.35)

Plot[g[v,2.7,0.35],{v,1.2,10},AxesOrigin -> {0.,-10.2}, TextStyle->{FontFamily->"Times",FontSize->14},AxesLabel-> TraditionalForm/@{v,g},PlotStyle->{ Thickness[0.005]}
, AxesStyle-> Thickness[0.004]]

g(v;T,p) = f(v,T) + p v  para T< (Subscript[T, c]) y p= (Subscript[p, max])  (temperatura t =2.7, presión p = pmax)

Plot[g[v,2.7,pmax],{v,1.2,10},AxesOrigin -> {0.,-10.95}, TextStyle->{FontFamily->"Times",FontSize->14},AxesLabel-> TraditionalForm/@{v,g},PlotStyle->{ Thickness[0.005]}
, AxesStyle-> Thickness[0.004]]

g(v;T,p) = f(v,T) + p v  para T< (Subscript[T, c]) y (Subscript[p, max])>p > (Subscript[p, coex])  (temperatura t =2.7, presión p = 0.265)

Plot[g[v,2.7,0.265],{v,1.2,10},AxesOrigin -> {0.,-11.025}, TextStyle->{FontFamily->"Times",FontSize->14},AxesLabel-> TraditionalForm/@{v,g},PlotStyle->{ Thickness[0.005]}
, AxesStyle-> Thickness[0.004]]

g(v;T,p) = f(v,T) + p v  para T< (Subscript[T, c]) y  p =(Subscript[p, coex])  (temperatura t =2.7, presión p = pcoex)

Plot[g[v,2.7,pcoex],{v,1.2,10},AxesOrigin -> {0.,-11.1}, TextStyle->{FontFamily->"Times",FontSize->14},AxesLabel-> TraditionalForm/@{v,g},PlotStyle->{ Thickness[0.005]}
, AxesStyle-> Thickness[0.004]]

g(v;T,p) = f(v,T) + p v  para T< (Subscript[T, c]) y (Subscript[p, coex])>p > (Subscript[p, min])  (temperatura t =2.7, presión p = 0.245)

Plot[g[v,2.7,0.245],{v,1.2,10},AxesOrigin -> {0.,-11.1}, TextStyle->{FontFamily->"Times",FontSize->14},AxesLabel-> TraditionalForm/@{v,g},PlotStyle->{ Thickness[0.005]}
, AxesStyle-> Thickness[0.004]]

p[2.194261904476178,2.7]

g(v;T,p) = f(v,T) + p v  para T< (Subscript[T, c]) y (Subscript[p, min])> p  (temperatura t =2.7, presión p = 0.14)

Plot[g[v,2.7,0.14],{v,1.2,25},AxesOrigin -> {0.,-10.5}, TextStyle->{FontFamily->"Times",FontSize->14},AxesLabel-> TraditionalForm/@{v,g},PlotStyle->{ Thickness[0.005]}
, AxesStyle-> Thickness[0.004]]