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 λmax = hc/5kT. At what wavelength does the maximum in ρ(λ,T) occur for T = 450, 1500, and 4500 K?

$$T = 450 \text{ K} \quad \Rightarrow \quad \lambda_{\text{max}} = {{(6.626 \times 10^{-34} \text{ J·s})(2.998 \times 10^8 \text{ m/s})} \over {5(1.381 \times 10^{-23} \text{ J/K})(450 \text{ K})}} = 6.39\ \mu\text{m}$$

$$T = 1500 \text{ K} \quad \Rightarrow \quad \lambda_{\text{max}} = {{(6.626 \times 10^{-34} \text{ J·s})(2.998 \times 10^8 \text{ m/s})} \over {5(1.381 \times 10^{-23} \text{ J/K})(1500 \text{ K})}} = 1.92\ \mu\text{m}$$

$$T = 4500 \text{ K} \quad \Rightarrow \quad \lambda_{\text{max}} = {{(6.626 \times 10^{-34} \text{ J·s})(2.998 \times 10^8 \text{ m/s})} \over {5(1.381 \times 10^{-23} \text{ J/K})(4500 \text{ K})}} = 639 \text{ nm}$$


 * 1) For a monatomic gas, one measure of the "average speed" of the atoms is the root mean square speed, vrms = &lt;v2&gt;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.

$$T = 100 \text{ K} \quad \Rightarrow \quad v_\text{He} = \sqrt{3(1.381 \times 10^{-23} \text{ J/K})(100 \text{ K}) \over 6.649 \times 10^{-27} \text{ kg}} = 789 \text{ m/s} \quad \Rightarrow \quad \lambda_\text{He} = $$