text: # Chapter 5 : Diffusion¶
  
  ### Example 5.1 Page No. 114¶
  
  In [3]:
  Ca=1.2       #Concentration at A in  kg/m**3
  Cb=0.8       #Concentration at B in  kg/m**3
  xa=5*10**-3   #Position 1 in m
  xb=10*10**-3  #Position 2 in m
  
  D=3*10**-11   #Diffusion coefficient in m**2/s
  J=-D*(Ca-Cb)/(xa-xb)
  
  print"Diffusion flux is ",J,"kg/m**2-s"
  
  Diffusion flux is  2.4e-09 kg/m**2-s
  
  
  ### Example 5.2 Page No. 117¶
  
  In [15]:
  from scipy.optimize import fsolve
  Co=0.25        #Initial Conc. in wt%
  Cs=1.2         #Surface conc. in wt%
  Cx=0.8         #Conc. at any x  in wt%
  x=5*10**-4     #Position in m
  D=1.6*10**-11  #Diffusion coeff in m**2/s
  
  import math
  C=1-((Cx-Co)/(Cs-Co))
  
  def f(z):
  return(0.4210-math.erf(z))
  z=fsolve(f,1)
  t=x**2/(4.0*D*z**2.0)
  
  print"Time required is ",round(t/3600.0,1),"h"
  
  Time required is  7.0 h
  
  
  ### Example 5.3 Page No. 118¶
  
  In [6]:
  D500=4.8*10**-14     #Diffusion coefficient at 500 C
  D600=5.3*10**-13     #Diffusion coefficient at 600 C
  t600=10             #Time in hours to diffuse
  
  t500=D600*t600/D500
  
  print"Time to diffuse at 500 C is ",round(t500,1),"h"
  
  Time to diffuse at 500 C is  110.4 h
  
  
  ### Example 5.4 Page No. 121¶
  
  In [8]:
  T=550+273            #in K
  D0=1.2*10**-4        #Temperature independent preexponential in m**2/s
  Qd=131000            #Activation energy in J/mol-K
  R=8.31               #Universal Gas constt
  
  import math
  D=D0*math.exp(-Qd/(R*T))
  
  print"Diffusion coefficient is ",round(D,14),"m**2/s"
  
  Diffusion coefficient is  5.8e-13 m**2/s
  
  
  ### Example 5.5 Page No.121¶
  
  In [15]:
  inv_T1=0.8*10**-3          #Reciprocal of temp.  in K**-1
  inv_T2=1.1*10**-3          #Reciprocal of temp.  in K**-1
  logD1=-12.4
  logD2=-15.45
  R=8.31                   #Gas law Constant in J/mol-K
  
  Qd=-2.3*R*(logD1-logD2)/(inv_T1-inv_T2)
  print"Activation energy is",round(Qd/1000,0),"KJ"
  
  D0=10**(logD2+(Qd*inv_T2/(2.3*R)))
  print"Preexponential factor is",round(D0,6),"m**2/s"
  
  Activation energy is 194.0 KJ
  Preexponential factor is 5.4e-05 m**2/s
  
  
  ### Design Example 5.1, Page No.122¶
  
  In [23]:
  C0=0.2           #Initial concentration in wt%
  Cs=1             #Surface conc in wt%
  Cx=0.6           #Conc at any position X in wt%
  x=7.5*10**-4      #Position in m
  D0=2.3*10**-5     #Preexponential factor in m**2/s
  R=8.31           #Gas law constant in J/mol-K
  Qd=148000        #Activation energy in J/mol
  
  C=1-((Cx-C0)/(Cs-C0))
  z=0.4747
  Dt=(x/(2*z))**2
  
  D=Dt/D0
  
  T1=900.0
  T2=950.0
  T3=1000.0
  T4=1050.0
  t1=D/math.exp(-Qd/(R*(T1+273)))/3600.0
  t2=D/math.exp(-Qd/(R*(T2+273)))/3600.0
  t3=D/math.exp(-Qd/(R*(T3+273)))/3600.0
  t4=D/math.exp(-Qd/(R*(T4+273)))/3600.0
  
  print"Temperature in Celcius  are",T1,T2,T3,T4
  print"Time is respectively ",round(t1,1),"h,",round(t2,1),"h,",round(t3,1),"h,",round(t4,1),"h"
  
  Temperature in Celcius  are 900.0 950.0 1000.0 1050.0
  Time is respectively  29.6 h, 15.9 h, 9.0 h, 5.3 h
  
  In [ ]:
explanation: The document contains multiple examples of math problems related to diffusion, each with a question and a corresponding answer. The questions are presented as code examples, and the answers are the results of the computations.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Chapter 5 : Diffusion¶
  
  ### Example 5.1 Page No. 114¶
  
  In [3]:
  Ca=1.2       #Concentration at A in  kg/m**3
  Cb=0.8       #Concentration at B in  kg/m**3
  xa=5*10**-3   #Position 1 in m
  xb=10*10**-3  #Position 2 in m
  
  D=3*10**-11   #Diffusion coefficient in m**2/s
  J=-D*(Ca-Cb)/(xa-xb)
  
  print"Diffusion flux is ",J,"kg/m**2-s"
  
  Diffusion flux is  2.4e-09 kg/m**2-s
  
  
  ### Example 5.2 Page No. 117¶
  
  In [15]:
  from scipy.optimize import fsolve
  Co=0.25        #Initial Conc. in wt%
  Cs=1.2         #Surface conc. in wt%
  Cx=0.8         #Conc. at any x  in wt%
  x=5*10**-4     #Position in m
  D=1.6*10**-11  #Diffusion coeff in m**2/s
  
  import math
  C=1-((Cx-Co)/(Cs-Co))
  
  def f(z):
  return(0.4210-math.erf(z))
  z=fsolve(f,1)
  t=x**2/(4.0*D*z**2.0)
  
  print"Time required is ",round(t/3600.0,1),"h"
  
  Time required is  7.0 h
  
  
  ### Example 5.3 Page No. 118¶
  
  In [6]:
  D500=4.8*10**-14     #Diffusion coefficient at 500 C
  D600=5.3*10**-13     #Diffusion coefficient at 600 C
  t600=10             #Time in hours to diffuse
  
  t500=D600*t600/D500
  
  print"Time to diffuse at 500 C is ",round(t500,1),"h"
  
  Time to diffuse at 500 C is  110.4 h
  
  
  ### Example 5.4 Page No. 121¶
  
  In [8]:
  T=550+273            #in K
  D0=1.2*10**-4        #Temperature independent preexponential in m**2/s
  Qd=131000            #Activation energy in J/mol-K
  R=8.31               #Universal Gas constt
  
  import math
  D=D0*math.exp(-Qd/(R*T))
  
  print"Diffusion coefficient is ",round(D,14),"m**2/s"
  
  Diffusion coefficient is  5.8e-13 m**2/s
  
  
  ### Example 5.5 Page No.121¶
  
  In [15]:
  inv_T1=0.8*10**-3          #Reciprocal of temp.  in K**-1
  inv_T2=1.1*10**-3          #Reciprocal of temp.  in K**-1
  logD1=-12.4
  logD2=-15.45
  R=8.31                   #Gas law Constant in J/mol-K
  
  Qd=-2.3*R*(logD1-logD2)/(inv_T1-inv_T2)
  print"Activation energy is",round(Qd/1000,0),"KJ"
  
  D0=10**(logD2+(Qd*inv_T2/(2.3*R)))
  print"Preexponential factor is",round(D0,6),"m**2/s"
  
  Activation energy is 194.0 KJ
  Preexponential factor is 5.4e-05 m**2/s
  
  
  ### Design Example 5.1, Page No.122¶
  
  In [23]:
  C0=0.2           #Initial concentration in wt%
  Cs=1             #Surface conc in wt%
  Cx=0.6           #Conc at any position X in wt%
  x=7.5*10**-4      #Position in m
  D0=2.3*10**-5     #Preexponential factor in m**2/s
  R=8.31           #Gas law constant in J/mol-K
  Qd=148000        #Activation energy in J/mol
  
  C=1-((Cx-C0)/(Cs-C0))
  z=0.4747
  Dt=(x/(2*z))**2
  
  D=Dt/D0
  
  T1=900.0
  T2=950.0
  T3=1000.0
  T4=1050.0
  t1=D/math.exp(-Qd/(R*(T1+273)))/3600.0
  t2=D/math.exp(-Qd/(R*(T2+273)))/3600.0
  t3=D/math.exp(-Qd/(R*(T3+273)))/3600.0
  t4=D/math.exp(-Qd/(R*(T4+273)))/3600.0
  
  print"Temperature in Celcius  are",T1,T2,T3,T4
  print"Time is respectively ",round(t1,1),"h,",round(t2,1),"h,",round(t3,1),"h,",round(t4,1),"h"
  
  Temperature in Celcius  are 900.0 950.0 1000.0 1050.0
  Time is respectively  29.6 h, 15.9 h, 9.0 h, 5.3 h
  
  In [ ]:
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains multiple examples of math problems related to diffusion, each with a question and a corresponding answer. The questions are presented as code examples, and the answers are the results of the computations.
  
  Label: multiple-question-and-answer