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The concentration profile of a chemical at a location $x$ and time $t$, denoted by $c(x,t)$, changes as per the following equation,

$$c(x,t)=\frac{c_0}{\sqrt{2\pi Dt}}\exp[-\frac{x^2}{2Dt}]$$

where $D$ and $c_0$ are assumed to be constant. Which of the following is correct?

  1. $\frac{\partial c}{\partial t}=D\frac{\partial^2 c}{\partial x^2}$
  2. $\frac{\partial c}{\partial t}=\frac{D}{2}\frac{\partial^2 c}{\partial x^2}$
  3. $\frac{\partial^2 c}{\partial t^2}=D\frac{\partial^2 c}{\partial x^2}$
  4. $\frac{\partial^2 c}{\partial t^2}=\frac{D}{2}\frac{\partial^2 c}{\partial x^2}$
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