Calculating the velocity of a molecule

[Victoire]

Hi, I'm stuck on a problem. I have this function that represents the velocity:
f(v)=v²e^(-mv2)/(2kT)

T is the temperature and equals 293.15K
m = 4.65 .10^-26 kg
k = 1.38 .10^-23 J/K

The problem wants us to find the derivative of the function. I found:
df/dv= (2v) e^(-mv2)/(2kT)+v²e^(-mv2)/(2kT)(-2mv/2kT)
= ve^(-mv2)/(2kT) (2- (mv²/kT))

I know the derivative I found is right, but now, I have to find the velocity for df/dv=0

We know v>0 and e^x>0
So, df/dv=0 => (mv²)/kT = 2
=> v= √(2kT/m)

That's where I'm stuck. The book tells me the answer is 1500km/h.
I don't understand this solution. I calculated many many times and didn't found the right solution. Can someone please help me?

 

[mjc123]

What answer did you get? Did you get 417? What are the units of that?
If you got a different answer, can you show us exactly how you got it? Your formula looks right.

 

[Victoire]

What answer did you get? Did you get 417? What are the units of that?
If you got a different answer, can you show us exactly how you got it? Your formula looks right.

Yes, I got 417!
I just plugged in all the values in the last equation that I wrote, v= √(2kT/m).
Since k is in J/K, T in K and m in kg, the unit of v is J/kg.
J also equals kg m² s^-2 so I assume the unit of v can also be m²s^-2?

 

[Victoire]

Oh, I got it! My answer is then in m.s^-1 and I just need to multiply by 3.6.
417*3.6=1501 km.h^-1

Thank you for the help :)

 

[mjc123]

Can't emphasise enough the importance of thinking UNITS, UNITS, UNITS; ALWAYS, ALWAYS, ALWAYS! It will save you a lot of trouble and avoid a lot of mistakes, both with homework questions like this and real problems in doing science.
For example, if you are asked for a velocity, the answer CANNOT be (e.g.) 417. It must be 417 SOMETHING - m/s, km/h, furlongs per fortnight or whatever. If you did that you would immediately see that your answer was not necessarily wrong, but another step was needed.