We used Dataplot to fit the C/C rational function model with the following 7 subset points to generate the starting values.

 TEMP        THERMEXP
 ----        --------
  10             0
  30             2
  40             3
  50             5
 120            12
 200            15
 800            20
       

EXACT RATIONAL FUNCTION FIT
NUMBER OF POINTS IN FIRST SET    =        7
DEGREE OF NUMERATOR              =        3
DEGREE OF DENOMINATOR            =        3
  
NUMERATOR  --A0  A1  A2  A3      =
   -0.2322993E+01  0.3528976E+00 -0.1382551E-01  0.1765684E-03
DENOMINATOR--B0  B1  B2  B3      = 
    0.1000000E+01 -0.3394208E-01  0.1099545E-03  0.7905308E-05
  
APPLICATION OF EXACT-FIT COEFFICIENTS
TO SECOND PAIR OF VARIABLES--
  
NUMBER OF POINTS IN SECOND SET           =      236
NUMBER OF ESTIMATED COEFFICIENTS         =        7
RESIDUAL DEGREES OF FREEDOM              =      229
  
RESIDUAL SUM OF SQUARES                  =  0.78246452E+02
RESIDUAL STANDARD DEVIATION (DENOM=N-P)  =  0.58454049E+00
AVERAGE ABSOLUTE RESIDUAL   (DENOM=N)    =  0.46998626E+00
LARGEST (IN MAGNITUDE) POSITIVE RESIDUAL =  0.95733070E+00
LARGEST (IN MAGNITUDE) NEGATIVE RESIDUAL = -0.13497944E+01
LARGEST (IN MAGNITUDE) ABSOLUTE RESIDUAL =  0.13497944E+01
  
  
       

LEAST SQUARES NON-LINEAR FIT
SAMPLE SIZE N =      236
MODEL--THERMEXP =(A0+A1*TEMP+A2*TEMP**2+A3*TEMP**3)/
                 (1+B1*TEMP+B2*TEMP**2+B3*TEMP**3)
REPLICATION CASE
REPLICATION STANDARD DEVIATION =     0.8131711930D-01
REPLICATION DEGREES OF FREEDOM =           1
NUMBER OF DISTINCT SUBSETS     =         235
  
FINAL PARAMETER ESTIMATES           (APPROX. ST. DEV.)    T VALUE
1  A0                   1.07913       (0.1710    )          6.3
2  A1                 -0.122801       (0.1203E-01)         -10.
3  A2                  0.408837E-02   (0.2252E-03)          18.
4  A3                 -0.142848E-05   (0.2610E-06)         -5.5
5  B1                 -0.576111E-02   (0.2468E-03)         -23.
6  B2                  0.240629E-03   (0.1060E-04)          23.
7  B3                 -0.123254E-06   (0.1217E-07)         -10.

RESIDUAL    STANDARD DEVIATION =         0.0818038210
RESIDUAL    DEGREES OF FREEDOM =         229
REPLICATION STANDARD DEVIATION =         0.0813171193
REPLICATION DEGREES OF FREEDOM =           1
LACK OF FIT F RATIO =       1.0121 = THE  32.1265% POINT OF THE
F DISTRIBUTION WITH    228 AND      1 DEGREES OF FREEDOM

       

Looking at the fitted function with the raw data appears to show a reasonable fit.

The 6-plot indicates no significant violation of the model assumptions. That is, the residuals have constant location and scale (from the residual plot in row 1, column 2), seem to be random (from the lag plot in row 2, column 1), and approximated well by a normal distribution (from the histogram and normal probability plots in row 2, columns 2 and 3).

The full size residual plot shows that the assumptions of constant location and scale for the residuals are valid. No distinguishing pattern is evident in the residuals.