GATE (TF) Textile 2021 Question Paper Solution | GATE/2021/TF/45

Question 45 (Textile Technology & Fibre Science)

The molecular weight (M) of a polymer is determined from Mark-Houwink Equation by using coefficient K = 11.5×10⁻³ ml/g and exponent a = 0.73. If the measured intrinsic viscosity [η] of the solution is 6.0×10² ml/g then the value of M×10⁻⁶, (rounded off to two decimal places), is __2.70 to 3.10__.

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Formula:
=KM^a ,where is viscosity , K is coefficient, M is molecular weight of polymer ,a is constant

As given in the question,
K=11.5×10^-3 ml/g,a=0.73 and =6.0×10² ml/g
By putting these values in to the formula-
=KM^a
6×10²=11.5×10^-3 xM^0.73
52173.913=M^0.73
M^0.73=52173.913

By taking log both side-
log_e(M)^0.73=log_e(52173.913)
0.73log_e(M)=10.86,
log_e(M)=14.88
M=e^14.88 , (By logarithm rule, log_e m ⁿ=m log_e n and log_e m=n to m=eⁿ)
M=2899358.3 ,
M=2.89×10⁶
M=2.89/10^-6
Mx10^-6=2.89 (Answer)

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