Molecular Structure and Chemical Bonding

Gibbs energy and equilibrium constant

The Gibbs energy G = H - TS combines enthalpy and entropy into one function that decides spontaneity at fixed temperature and pressure. For a process, \Delta G = \Delta H - T\Delta S. At constant T and P, a process is spontaneous when \Delta G < 0, and the system is at equilibrium when \Delta G = 0. For a reaction in its standard state, \Delta G^\circ = -RT\ln K = -2.303\,RT\log K, so a measured equilibrium constant gives the standard Gibbs energy directly. Rearranging \Delta G^\circ = \Delta H^\circ - T\Delta S^\circ gives \Delta S^\circ = (\Delta H^\circ - \Delta G^\circ)/T, which is how an entropy change is recovered from \log K and \Delta H^\circ at a given temperature.

The criterion for spontaneity depends on which variables are held constant. Each condition pairs with a different state function:

• Constant T and P: \Delta G < 0.
• Constant T and V: \Delta A < 0, where A = U - TS is the Helmholtz energy.
• Constant U and V (isolated system): \Delta S > 0.
• Constant S and V: \Delta U \le 0.

A frequent mistake is to write \Delta U > 0 or \Delta G > 0 as a spontaneity condition. Spontaneous changes always lower the relevant potential, not raise it. Careful attention to the fixed variables is what separates the correct statements from the incorrect ones.

For an electrochemical cell, \Delta_rG = -nFE, where n is the number of electrons transferred in the balanced cell reaction, F = 96500 C mol^{-1}, and E is the cell potential. The temperature coefficient of the cell potential gives the entropy change, \Delta_rS = nF\left(\partial E/\partial T\right)_P, and the enthalpy then follows from \Delta_rH = \Delta_rG + T\Delta_rS. The sign of the temperature coefficient fixes the sign of \Delta_rS, and n must be counted from the balanced equation rather than assumed.

Overlap integral of hydrogen 1s orbitals

The overlap integral S = \langle\phi_A|\phi_B\rangle = \int \phi_A^*\,\phi_B\,d\tau measures how much two atomic orbitals occupy the same region of space. For normalised functions, the Cauchy–Schwarz inequality gives |S| \le 1, and S equals 1 only when the two functions are identical. For two hydrogen 1s orbitals on nuclei separated by a distance R, the exact result is S = e^{-\rho}\left(1 + \rho + \rho^2/3\right) with \rho = R/a_0, where a_0 is the Bohr radius.

This function decreases monotonically from S = 1 at zero separation toward S = 0 as R \to \infty. Because the orbitals are real, S is real, and it stays between these two limits at every separation. Therefore a statement that is true for all separations is usually a bound such as 0 \le S \le 1, while an exact value of 0 or 1 holds only at a limit (coincident nuclei or infinite separation). Statements that claim a specific value or an imaginary value for S are wrong for real hydrogen 1s orbitals.

Crystal system of unit cells

A crystal system is defined by the relations among the three cell edge lengths a, b, c and the three interaxial angles \alpha, \beta, \gamma. The seven systems are:

• Cubic: a = b = c, all angles 90^\circ.
• Tetragonal: a = b \ne c, all angles 90^\circ.
• Orthorhombic: a \ne b \ne c, all angles 90^\circ.
• Monoclinic: a \ne b \ne c, \alpha = \gamma = 90^\circ \ne \beta.
• Triclinic: a \ne b \ne c, no angle equal to 90^\circ in general.
• Hexagonal: a = b \ne c, \gamma = 120^\circ, \alpha = \beta = 90^\circ.
• Trigonal (rhombohedral): a = b = c, \alpha = \beta = \gamma \ne 90^\circ.

When a problem gives planar sections of a unit cell, first check the shape of each section. A rectangular section shows that the two edges in that plane are perpendicular. If all three sections are rectangles, all angles are 90^\circ, and then the system is orthorhombic when no two edges are equal, tetragonal when exactly two are equal, and cubic when all three are equal. Edge lengths are obtained from the intercepts of the planes, and the consistency of the three planes should be checked before the answer is chosen.

Mark–Houwink viscosity–molecular weight relation

The intrinsic viscosity [\eta] of a dilute polymer solution is the limiting value of the reduced viscosity as the concentration goes to zero, and it reflects the hydrodynamic size of a single polymer coil. The Mark–Houwink equation relates it to the molecular weight M by [\eta] = K M^{a}. The constants K and a depend on the specific polymer, solvent and temperature, and they must be taken from the data given for that system.

The exponent a describes the solvent quality. Values near 0.5 correspond to a theta solvent, where the chain behaves ideally, while values approaching 0.8 indicate a good solvent in which the coil is more expanded. Rigid rod-like polymers give larger values. Because the equation is a power law, the molecular weight is found by inverting it, M = \left([\eta]/K\right)^{1/a}. The result is a viscosity-average molecular weight, not a number-average value.

When using the equation, the units of K must match the units of [\eta] and the molar mass unit must follow from them. A common error is to forget that the exponent 1/a is larger than 1, so small relative errors in [\eta]/K are amplified in M. Unit conversion should be done before the power is taken.

Thermogravimetric mass loss analysis

Thermogravimetric analysis (TGA) records the mass of a sample as it is heated at a controlled rate. A mass step on the curve shows that a volatile species has been released, for example the loss of water of hydration, the decomposition of a compound, or the escape of a gas. Under an inert atmosphere such as nitrogen, only the loss of volatile products is seen, whereas in air an oxidation step may increase the mass. The atmosphere must therefore be read from the question before the mass change is interpreted.