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## Homework Statement

In the chemical reaction problem, assume that the reaction is C + C --> Product, and the chemical reaction rate is [itex]r=kc^{2}[/itex], where k is the reaction constant. What is the dimension of k? Define dimensionless variables and reformulate the problem in dimensionless form. Solve the dimensionless problem to determine the concentration.

## Homework Equations

We are assuming temperature is not relevant.

The mathematical model given by the initial value of the problem is as follows.

[itex]\frac{dc}{dt} = \frac{q}{V}(c_{i}-c)-k^{2}c , t>0 [/itex]

In this problem, c is the concentration, t is time, q is the inflow rate (mass/time), V is the volume, and k is the reaction constant.

## The Attempt at a Solution

First, I equalized the dimensions on both sides to get that [itex]k=\frac{V}{MT}[/itex] . This part seems simple as the dimensions work now.

However, I am being asked to dimensionalize the problem. Finding a dimensionless concentration is easy, [itex]C=\frac{c}{c_{i}}[/itex]. I am trying to find a dimensionless time. I have gotten as far as [itex]\tau=\LARGE{\frac{t}{\frac{1}{c_{0}k}}}[/itex], since [itex]c_{0} * k[/itex] has units of [itex]c_{0}=\frac{M}{V}[/itex], and [itex]k=\frac{V}{MT}[/itex] This much is easy. However I look in the solution manual. It is telling me that the dimensionless time [itex]\tau=\LARGE{\frac{t}{\frac{V}{Q}}}[/itex]. When I work out the units, I get [itex]t[/itex] on the top, but the bottom is [itex]\LARGE{\frac{V}{Q}=\frac{V}{\frac{M}{T}}=T\frac{M}{V}}[/itex]. But [itex]\frac{M}{V}[/itex] is not dimensionless, so [itex]\tau=\frac{M}{V}[/itex]. Can anyone figure this out? Seems beyond obvious and ridiculously simple of a problem, is my solution manual blatantly wrong or am I just completely clueless?

Thank you!!