Penetration theory states that the mass transfer co-coefficient is equal to (where, D_e is diffusivity and 't' is time)

ANS:C - (4D_e/πt)^1/2

The Penetration theory states that the mass transfer coefficient (𝐾K) is proportional to the inverse square root of the diffusion time (𝑡t). Mathematically, this relationship is represented as: 𝐾∝1𝑡K∝t​1​ The diffusivity (𝐷𝑒De​) is related to the diffusion time by the equation: 𝐷𝑒=𝑥24𝑡De​=4tx2​ where 𝑥x is the distance traveled by the diffusing species. Therefore, substituting the expression for 𝐷𝑒De​ into the relationship for 𝐾K, we get: 𝐾∝𝑥24𝑡K∝4tx2​​ 𝐾∝𝑥2𝑡K∝2t​x​ However, since the proportionality constant is typically determined empirically, it is common to see the equation written as: 𝐾=𝑥2𝑡K=2t​x​ So, the correct option from the provided choices would be: (4𝐷𝑒𝜋𝑡)1/2(πt4De​​)1/2