Solving Radiative Cooling Rate Homework Equation

[Piano man]

Homework Statement

The first part of this question deals with interstellar gas collisions at a certain velocity, and the resultant temperature of the cloud is obtained.

Then this question:
The main coolant has the following radiative loss rate
\Lambda=n_H10^{-28}T_{gas} \text{erg cm}^{-3}\text{s}^{-1}

(i)Write down the equation for the kinetic temperature for the gas.
(ii)How long does it take for the cloud to reach half of the initial impact temperature if the volume remains the same?

Homework Equations

I've found this equation for cooling time
t_c=\frac{3kT_s}{n_H\Lambda(T_s)}
where Ts is the post-shock temperature.

The Attempt at a Solution

The first part is simple enough, just
KE=\frac{1}{2}mv^2=\frac{3}{2}kT

But when I sub in the given value for Lambda into the cooling time equation, the temperatures cancel, leaving
t_c=\frac{3kT_s}{n_H(n_H10^{-28}T_{s})}=\frac{3k}{n_H^2 10^{-28}}

So if I'm looking for the time taken to cool to Ts/2, I have nowhere to sub in, as the cooling time is now independent of temperature.
Where have I gone wrong?

 

[Piano man]

Any ideas?