Hi Derek,

I did some syntax correction and divided your code into 3 sections. The first establishes initial values for the iterations. The second performs the iterations 10 times and accumulates the Y1 results. It shows that the iterations quickly take Y1 to a stable value greater than the termination condition. The third it the while loop, but of course it does not terminate. I don't know enough of the fundamentals to suggest more. perhaps other initial conditions would behave differently, but maybe not.

Kind regards, David

PS. In Mathematica, it is good to use variable names starting with lower case, since reserved words all begin with upper.

In[1]:= (* set up initial values *)
gamma = 21/28.96;
P = 2000;
T = 581.67;
Tpc = 168 + 325*gamma - 12.5*gamma^2;
Ppc = 667 + 15*gamma - 37.5*gamma^2;
Tpr = T/Tpc;
t = 1/Tpr;
Ppr = 2000/Ppc;
X1 = -0.06125*Ppr*t*Exp[-1.2*(1 - t)^2];
X2 = (14.76*t - 9.76*t^2 + 4.58*t^3);
X3 = (90.7*t - 242.2*t^2 + 42.4*t^3);
X4 = (2.18 + 2.82*t);
Y = 0.0125*Ppr*t*Exp[-1.2*(1 - t)^2];
F = X1 + (Y + Y^2 + Y^3 + Y^4)/(1 - Y)^3 - X2*Y^2 + X3*Y^X4;
Fpr = (1 + 4*Y + 4*Y^2 - 4*Y^3 + Y^4)/(1 - Y)^4 - 2*X2*Y + 
   X3*X4*Y^(X4 - 1);
Y1 = Y - F/Fpr;

In[17]:= (* do the first 10 iterations and accumulate Y1 in a list *)

y1Values = {};
Do[
 Y = Y1; 
 F = X1 + (Y + Y^2 + Y^3 + Y^4)/(1 - Y)^3 - X2*Y^2 + X3*Y^X4;
 Fpr = (1 + 4*Y + 4*Y^2 - 4*Y^3 + Y^4)/(1 - Y)^4 - 2*X2*Y + 
   X3*X4*Y^(X4 - 1);
 Y1 = Y - F/Fpr;
 AppendTo[y1Values, Y1],
 {10}
 ]

In[19]:= (* appears to stagnate with 0.147417 a stable value for y1 *)
\
    y1Values

Out[19]= {0.14882, 0.147323, 0.147424, 0.147417, 0.147417, 0.147417, \
0.147417, 0.147417, 0.147417, 0.147417}

In[20]:= (* the While will not terminate *)
While[Abs[Y1] > 10^-4,
  Y = Y1; 
  F = X1 + (Y + Y^2 + Y^3 + Y^4)/(1 - Y)^3 - X2*Y^2 + X3*Y^X4;
   Fpr = (1 + 4*Y + 4*Y^2 - 4*Y^3 + Y^4)/(1 - Y)^4 - 2*X2*Y + 
    X3*X4*Y^(X4 - 1) ;
  Y1 = Y - F/Fpr
  ];

Out[20]= $Aborted

As Daniel says, it would be a big help if you would post code instead of an image. You can find instructions on posting here: How to post

Also, your algorithm appears to be trying to accomplish a fit to some function. You might tell us more about this at the top level, since Mathematica very often has high level functions which serve such purposes better than procedural programming. In my experience, For, Do, and While are seldom the best construct in Wolfram code.

Kind regards, David

Inside the while loop, you probably want to replace the double equal signs with single equal signs and put the semicolon and the end of those lines.

What you show is an image, and that cannot be cut and pasted into Mathematica. SO it's not really possible for anyone to experiment with the code.