g

Examples of g in the following topics:

• Non-Thermal Emission

  • $\displaystyle \frac{n_2 g_1}{n_1 g_2} = \exp \left (-\frac{h\nu}{kT} \right ) < 1$

• Differentiation Rules

  • $(\alpha f + \beta g)' = \alpha f' + \beta g'$
  • $\displaystyle{\left ( \frac {f}{g} \right )' = \frac {f'g - fg'}{g^2}}$
  • for all functions $f$ and $g$ at all inputs where $g \neq 0$.

• Composition of Functions and Decomposing a Function

  • We read the left-hand side as "$f$" composed with $g$ at $x$, and the right-hand side as  "$f$ of $g$ of $x$."  
  • In general, $(f∘g)$ and $(g∘f)$ are different functions.
  • In other words, in many cases$f(g(x))≠g(f(x))$ for all $x$. 
  • To evaluate $f(g(3))$, first substitute, or input the value of $3$ into $g(x)$ and find the output.  
  • For $g(f(x))$, input the $f(x)$ function, $-2x$ into the $g(x)$ function, and then simplify:

• Attachment and Entry of Herpes Simplex

  • The genome encodes for 11 different glycoproteins, four of which, gB, gC, gD and gH, are involved in viral attachment.
  • Initial interactions occur when viral envelope glycoprotein C (gC) binds to a cell surface particle called heparan sulfate.
  • A second glycoprotein, glycoprotein D (gD), binds specifically to at least one of three known entry receptors.
  • Once bound to the HVEM, gD changes its conformation and interacts with viral glycoproteins H (gH) and L (gL), which form a complex.
  • Afterward, gB interaction with the gH/gL complex creates an entry pore for the viral capsid.

• Converting between Mass and Number of Moles

  • In a compound of NaOH, the molar mass of Na alone is 23 g/mol, the molar mass of O is 16 g/mol, and H is 1 g/mol.
  • $23 \space \text{g/mol} +16 \space \text{g/mol}+ 1 \space \text{g/mol} = 40 \space \text{g/mol}$
  • Since the molar mass of NaOH is 40 g/mol, we can divide the 90 g of NaOH by the molar mass (40 g/mol) to find the moles of NaOH.
  • $90 g \space \text{NaOH} \times \frac{1 \space mol}{40 g} = 2.25 \space \text{mol NaOH}$
  • $\frac{10.0 \ g\ CH_4}{0.623 \ mol\ CH_4} = 16.05 \ g/mol \ CH_4 $

• Free Energy Changes for Nonstandard States

  • The apparent similarity of the terms $\Delta G$, $\Delta G^{\circ}$ and $\Delta G_f^{\circ}$ makes it easy to overlook the distinctions between them; the similarity is a major source of confusion.
  • The red line on the left plots the values of $\Delta G$ for the $\Delta G^{\circ}$ > 0 reaction.
  • The plot on the right is for the $\Delta G^{\circ}$ < 0 reaction, for which $\Delta G^{\circ}$is shown at point 4 on the diagram.
  • What would happen if $\Delta G^{\circ}$ were 0?
  • Graph of $G(T,p) = f(\xi)$.

• Lagrange Multiplers

  • Maximize $f(x,y)$ subject to $g(x,y)=c$.
  • We can visualize contours of $f$ given by $f(x, y)=d$ for various values of $d$, and the contour of $g$ given by $g (x, y) = c$.
  • Suppose we walk along the contour line with $g = c$.
  • $\nabla_{x,y} f = - \lambda \nabla_{x,y} g$, where $\nabla_{x,y} f= \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right)$ and $\nabla_{x,y} g= \left( \frac{\partial g}{\partial x}, \frac{\partial g}{\partial y} \right)$ are the respective gradients.
  • Since both $g_x \neq 0$ and $g_y \neq 0$, the Lagrange multiplier $\lambda = 0$ at the minimum.

• Types of Social Movements

  • Social movements can be aimed at change on an individual level (e.g., AA) or change on a broader, group or even societal level (e.g., anti-globalization).
  • Social movements can also advocate for minor changes (e.g., tougher restrictions on drunk driving; see MADD) or radical changes (e.g., prohibition).

• Gas Stoichiometry

  • According to the above reaction, what volume of NO2(g) is produced from the combustion of 100 g of NH3(g), assuming the reaction takes place at standard temperature and pressure?
  • From the periodic table, we can determine that the molar mass of ammonia, NH3(g), is 17 g/mol, and perform the following stoichiometric calculation:
  • $\left(\frac{\text{100 g }NH_3}{ }\right)\times\left(\frac{\text{1 mol }NH_3}{\text{17 }g}\right)\times \left(\frac{\text{4 mol }NO_2}{\text{4 mol }NH_3}\right)\times \left(\frac{\text{22.4 }L}{\text{1 mol }NO_2}\right)=\text{132 L }NO_2(g)$

• B.8 Chapter 8

  • $\displaystyle \Delta E \sim \frac{g_p g_e}{8} \frac{e^2}{m c^2} \frac{m^3 e^6}{\hbar^4 M} = \frac{g_p g_e}{8} \frac{\alpha \hbar c}{m c^2} \frac{m^3 \alpha^3 \hbar^3 c^3}{\hbar^4 M} = \frac{g_p g_e}{8} \alpha^4 \frac{m}{M} m c^2 = 10^{-6}~\text{eV} $
  • $\displaystyle E = \frac{g_p g_e}{4} \frac{e^2}{m c^2} \frac{\hbar^2}{M r^3} {\bf s}_1 \cdot {\bf s}_2$
  • $\displaystyle E = \frac{g_p g_e}{4} \frac{e^2}{m c^2} \frac{m^3 e^6}{\hbar^4 M} = \frac{g_p g_e}{4} \frac{\alpha \hbar c}{m c^2} \frac{m^3 \alpha^3 \hbar^3 c^3}{\hbar^4 M} = \frac{g_p g_e}{4} \alpha^4 \frac{m}{M} m c^2 \left ({\bf s}_1 \cdot {\bf s}_2 \right ).$
  • $\displaystyle \Delta E_{F=0,F=1} = \frac{g_p g_e}{4} \alpha^4 \frac{m}{M} m c^2 = 2 \times 10^{-6} \text{eV}$
  • You may take $U(T)=g_0=2$ and $U^+(T)=g_0^+=2$.