![Q.36 The concentration profile of a chemical at a location x and time t, denoted by c(x,t), changes as per the following equation, c(x,t) =c0 /√(2πDt) exp[ −x2 / (2Dt)] where D and c0 are assumed to be constant. Which of the following is correct? ∂c/∂t = D ∂2c / ∂x2 ∂c/∂t = (D/2) ∂2c / ∂x2 ∂2c/∂t2 = D ∂2c / ∂x2 ∂2c/∂t2 = (D/2) ∂2c / ∂x2](https://www.letstalkacademy.com/wp-content/uploads/2026/01/slide_36-1.jpg)
Q.36 The concentration profile of a chemical at a location x and time t,
denoted by c(x,t), changes as per the following equation,
c(x,t) =c0 /√(2πDt) exp[ −x2 / (2Dt)]
where D and c0 are assumed to be constant.
Which of the following is correct?
- ∂c/∂t = D ∂2c / ∂x2
- ∂c/∂t = (D/2) ∂2c / ∂x2
- ∂2c/∂t2 = D ∂2c / ∂x2
- ∂2c/∂t2 = (D/2) ∂2c / ∂x2
Introduction
Diffusion processes in chemical and mass transfer systems are commonly described using
Fick’s second law. The Gaussian concentration profile is a classical analytical solution
of this law. In this article, we verify which differential equation is satisfied by the
given concentration profile.
Problem Statement
The concentration profile is given by:
c(x,t) = c0 /√(2πDt) exp ( -x2 / 2Dt )
where D and c0 are constants.
Step 1: Physical Interpretation
The given expression represents a Gaussian distribution, which is a well-known solution
of Fick’s second law of diffusion:
∂c / ∂t = D ∂2c / ∂x2
Step 2: Time Derivative
Differentiating c(x,t) with respect to time:
∂c / ∂t = c ( -1 / 2t + x2 / 2Dt2 )
Step 3: Spatial Derivative
Differentiating twice with respect to x:
∂2c / ∂x2 = c ( x2 / D2t2 – 1 / Dt )
Multiplying both sides by D:
D ∂2c / ∂x2 = c ( -1 / t + x2 / Dt2 )
Step 4: Comparison
Dividing the above expression by 2 gives:
∂c / ∂t = D ∂2c / ∂x2
Hence, the given concentration profile satisfies Fick’s second law.
Correct Answer
Option (A):
∂c / ∂t = D ∂2c / ∂x2
Conclusion
The given concentration profile is a classical Gaussian solution of Fick’s second law
of diffusion. Therefore, the correct governing equation is:
∂c / ∂t = D ∂2c / ∂x2
