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Question 125 pts An FCC nickel-carbon alloy initially containing 0.20 wt% carbon is carburized at an elevated temperature and in an atmosphere in which the surface carbon concentration is maintained at 1.0 wt%. Given: Do = 2.3 x 10-5 m2/s, Qd = 111,000 J/mol. If, after 49.5 h, the concentration of carbon is 0.35 wt% at a position 4.0 mm below the surface, determine the temperature at which the treatment was carried out. Boltzmann constant, k = 8.617x10-5 eV/(atom-K)= 1.38x10-23 J/(atom-K) and Avogadro’s number, NA=6.022×1023 atom/mol. Group of answer choices 1300 K 1375 K 975 K 1027 K

Sagot :

Answer:

975 K

Explanation:

Here, given :

The alloy is a FCC nickel-carbon.

[tex]$D_o= 2.3 \times 10^{-5} \ m^2/s$[/tex]

[tex]$Q_d=111,000 \ J/mol$[/tex]

Boltzmann Constant, [tex]$k=8.617 \times 10^{-5} \ eV/(atom-K)$[/tex]

Therefore,

[tex]$\frac{C_x-C_o}{C_s-C_o}= \frac{035-0.20}{1.0-0.20}$[/tex]

              = 0.1875

             [tex]$=1-\text{erf}\left(\frac{x}{2\sqrt{Dt}}\right)$[/tex]

So, [tex]$\text{erf}\left(\frac{x}{2\sqrt{Dt}}\right) = 0.8125$[/tex]

Therefore,

  w                  erf w

0.92             0.80677

 y                 0.8125

0.96             0.82542

Now,

[tex]$\frac{y-0.92}{0.96-0.92} = \frac{0.8125-0.80677}{0.82542-0.80677}$[/tex]

y = 0.93228

[tex]$\text{erf}\left(\frac{x}{2\sqrt{Dt}}\right) = 0.93228$[/tex]

[tex]$D=\frac{x^2}{4t(0.93228)^2}$[/tex]

[tex]$D=\frac{(4\times 10^{-3})^2}{4\times 49.5 \times 3600 \times (0.93228)^2}$[/tex]

   [tex]$= 2.58 \times 10^{-11}$[/tex]

[tex]$T=\frac{Q_d}{R(\ln D_o - \ln D)}$[/tex]

[tex]$T=\frac{111000}{8.31(\ln (2.3 \times 10^{-5}) - \ln (2.58 \times 10^{-11}))}$[/tex]

T = 974.84 K

T = 975 K

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