# Finding a dimensionless time in a very basic problem

## Homework Statement

In the chemical reaction problem, assume that the reaction is C + C --> Product, and the chemical reaction rate is $r=kc^{2}$, 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.

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

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 $k=\frac{V}{MT}$ . This part seems simple as the dimensions work now.

However, I am being asked to dimensionalize the problem. Finding a dimensionless concentration is easy, $C=\frac{c}{c_{i}}$. I am trying to find a dimensionless time. I have gotten as far as $\tau=\LARGE{\frac{t}{\frac{1}{c_{0}k}}}$, since $c_{0} * k$ has units of $c_{0}=\frac{M}{V}$, and $k=\frac{V}{MT}$ This much is easy. However I look in the solution manual. It is telling me that the dimensionless time $\tau=\LARGE{\frac{t}{\frac{V}{Q}}}$. When I work out the units, I get $t$ on the top, but the bottom is $\LARGE{\frac{V}{Q}=\frac{V}{\frac{M}{T}}=T\frac{M}{V}}$. But $\frac{M}{V}$ is not dimensionless, so $\tau=\frac{M}{V}$. 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!!

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