Problem 3.38 H C l has a rotational constant of 10.59 c m ? 1 and a vibrational

Question

                                             Problem 3.38                                                                                                                                  HCl has a rotational constant of 10.59 cm?1 and a vibrational constant of 2991                             cm?1.                                                                                                                              Part A                                                                   Find the temperature, if there is one, at which the J = 0                         and J = 1 rotational states have the same population.                                                                   Express your answer using two decimal places.                                                                                                                                                                                  
                                                                                             ________________________ K
                                                
                                                
                                                                                                                                                        
                                                                                                    Part B                                                       
                                                      Find the temperature, if there is one, at which the = 0 and = 1 states have the same population.                                                                                           
                                    27.74 K                                                                                                       
                                    0 K                                                                                                       
                                    55.48 K                                                                                                       
                                    There is no finite temperature                                              Problem 3.38                                                               Problem 3.38                                                                                                HCl has a rotational constant of 10.59 cm?1 and a vibrational constant of 2991                             cm?1.                                                                                                                     HCl has a rotational constant of 10.59 cm?1 and a vibrational constant of 2991                             cm?1.                                                                           HCl has a rotational constant of 10.59 cm?1 and a vibrational constant of 2991                             cm?1.                                                                       Part A                                                                   Find the temperature, if there is one, at which the J = 0                         and J = 1 rotational states have the same population.                                                                   Express your answer using two decimal places.                                                                                                                                                                                  
                                                                                             ________________________ K
                                                
                                                
                                                                                                                                                        
                                                                                       Part A                                              Find the temperature, if there is one, at which the J = 0                         and J = 1 rotational states have the same population.                                              Express your answer using two decimal places.                                                                                                                                                             
                                                                                             ________________________ K
                                                
                                                
                                                                                                                                                        
                                                                                                                                                            
                                                                                             ________________________ K
                                                
                                                
                                                                                                                                                                                    
                                                                                             ________________________ K
                                                
                                                
                                                                                                      
                                             Part B                                      
                                                                 
                                                                                             ________________________ K
                                                
                                                
                                            

Explanation / Answer

(A) The answer is: 27.74 K


Boltzmann distribution for rotational energy levels:

N(J)/N(0) = (2J + 1) exp[-(BhcJ(J + 1)/kT]

where N is population, B is rotational constant in cm-1, h is Planck constant, c is speed of light in cm/s, k

is Boltzmann constant and T is temperature


For J = 1 and N(1) = N(0)

N(1)/N(0) = (2 x 1 + 1) x exp[-10.59 x 6.626 x 10^(-34) x 2.998 x 10^10 x 1 x (1 + 1)/(1.3806 x 10^(-23) x T)]


1 = 3 x exp(-30.466/T)

-30.475/T = ln(1/3) = -1.0986

Temperature T = 27.74 K



(B) The answer is: There is no finite temperature


Boltzmann distribution for vibrational energy levels:

N(v)/N(0) = exp[-(E(v) - E(O))/kT]

where N is population, E is vibrational energy level, k is Boltzmann constant and T is temperature


For v = 1:

N(1)/N(0) = exp[-(E(1) - E(O))/kT]

= exp(-hcw/kT) (where w is vibrational constant)

= exp[-6.626 x 10^(-34) x 2.998 x 10^10 x 2991/(1.3806 x 10^(-23) x T)]

= exp(-4303.6/T) which is less than 1 for all positive values of T


Thus there is no finite temperature T for which N(1)/N(0) = 1

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