From my understanding plotting the conc. over time for a 0-order reaction yield a negative linear slope. Plotting conc vs time for a first order reaction intuitively yields an exponential type slope. however, my teacher never covered the looks of a conc vs time graph for a second order reaction. What may this look like. In conjunction, what does it look like when we graph its rate over concentration?
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Let's take a single-component second-order reaction:
$$\ce{2A -> B}$$
If it is a 2nd-order reaction, the rate will be proportional to the concentration of $A$, squared:
$$\frac{dA}{dt} = -kA^2$$
That equation can be integrated:
$$\frac{dA}{A^2} == -k dt$$
$$-(A^{-1}-A_0^{-1}) = -k(t-t_0)$$
And solving for $A$:
$$A = \frac{1}{k(t-t_0) + A_0^{-1}}$$
That is the equation of a hyberbola. It goes to zero more slowly than a the expression for a first-order reaction, i.e. an exponential decay. This makes sense because as $A$ gets more and more diluted as the reaction occurs, collisions between $A$ molecules become rarer and rarer.
See this web page for more information.