Show that the linear combinations \(h_{1}=\mathrm{s}+\mathrm{p}_{x}+\mathrm{p}_{y}+\mathrm{p}_{z}\) and \(h_{2}\) = \(\mathrm{s}-\mathrm{p}_{x}-\mathrm{p}_{y}+\mathrm{p}_{z}\) are mutually orthogonal.
Text Transcription:
h_1=s+p_x+p_y+p_z
h_2
s-p_x-p_y+p_z
Chapter 16 — Chemical Thermodynamics 1 I. spontaneity A. spontaneous process — once started, proceeds on its own without a continuous external influence 1. irreversible under same conditions at which it is spontaneous 2. examples: a) gas particles mixing by diffusing b) heat flowing from higher temperature to lower temperature c) ice melting above its freezing point d) batteries discharging e) oxidation of ion B. non spontaneous process — reverse of a spontaneous process under the same conditions 1. examples: a) gas particles returning to their original state b) heat flowing from lower temperature to a higher temperature c) ice melting below its freezing point d) batteries charging e) iron (III) oxide decomposing to iron and oxygen C. reaction is spontaneous in the forward direction and non spontaneous in the reverse direction D. spontaneity of a reaction or process depends on the value of ∆G: 1. ∆G = ∆H — T∆S 2. if ∆G < 0 (negative), process is spontaneous 3. if ∆G > 0 (positive), process is non spontaneous 4. if ∆G = 0, process is at equilibrium II. entropy — level of disorder in a system A. measured by: complexity of the molecule (mass, # of bonds, etc.) 1. 2. mixing — forming solutions (thing that is mixed is more random) 3. increasing
