Hello. Encountered the following question:

"Calculate the number of molecules in 1cm^3 of an ideal gas at 27C and a pressure of 10mm Hg. Mean kinetic energy of a molecule at 27C is 4*10^-14 erg; the density of mercury is 13.6gm/cm^3"

I tried solving this using PV = nRT (as it is applicable to ideal gases)

(10/760) atm * (1/1000)L = n * 0.0821 (L)(atm) * 300K ... [760mmHg = atmospheric pressure]

n = no. of moles ~= 5.34 * 10^(-7)

no. of molecules ~= 3.22 * 10^17

HOWEVER: the question is solved using the given mean KE value and using the relation - P = 2/3 KE

The final answer given is 4.9*10^17 molecules

My question is - why is my approach wrong?

Isn't the use of the ideal gas equation correct?

Thanks a lot.