User:Rainmonger/Quantum chemistry problems

1. The distribution in wavelengths of the light emitted from a radiating blackbody is a sensitive function of the temperature. This dependence is used to measure the temperature of hot objects, without making physical contact with those objects, in a technique called optical pyrometry. In the limit (hc/λkT) >> 1, the maximum in a plot of ρ(λ,T) versus λ is given by λ_max = hc/5kT. At what wavelength does the maximum in ρ(λ,T) occur for T = 450, 1500, and 4500 K?
    
   $T=450{\text{ K}}\quad \Rightarrow \quad \lambda _{\text{max}}={{(6.626\times 10^{-34}{\text{ J·s}})(2.998\times 10^{8}{\text{ m/s}})} \over {5(1.381\times 10^{-23}{\text{ J/K}})(450{\text{ K}})}}=6.39\ \mu {\text{m}}$
    
   $T=1500{\text{ K}}\quad \Rightarrow \quad \lambda _{\text{max}}={{(6.626\times 10^{-34}{\text{ J·s}})(2.998\times 10^{8}{\text{ m/s}})} \over {5(1.381\times 10^{-23}{\text{ J/K}})(1500{\text{ K}})}}=1.92\ \mu {\text{m}}$
    
   $T=4500{\text{ K}}\quad \Rightarrow \quad \lambda _{\text{max}}={{(6.626\times 10^{-34}{\text{ J·s}})(2.998\times 10^{8}{\text{ m/s}})} \over {5(1.381\times 10^{-23}{\text{ J/K}})(4500{\text{ K}})}}=639{\text{ nm}}$
    
   
2. For a monatomic gas, one measure of the "average speed" of the atoms is the root mean square speed, v_rms = <v²>^1/2 = ${\sqrt {3kT/m}}$, in which m is the molecular mass and k is the Boltzmann constant. Using this formula, calculate the de Broglie wavelength for He and Ar atoms at 100 and at 500 K.
    
   $T=100{\text{ K}}\quad \Rightarrow \quad v_{\text{He}}={\sqrt {3(1.381\times 10^{-23}{\text{ J/K}})(100{\text{ K}}) \over 6.649\times 10^{-27}{\text{ kg}}}}=789{\text{ m/s}}\quad \Rightarrow \quad \lambda _{\text{He}}=$