Energy and State Function

Energy change is independent of the pathway; however, work and heat are both dependent on the pathway. State Function - depens only on its present state Endothermic(吸熱) Exothermic(放熱)

First law of thermodynamics

The energy of the universe is constant

Internal Energy \(E\)

$$\Delta E = q + w $$

Under const. pressure, the work done on the system itself is negative to the work done on its surroundings.

$$w = - P\Delta V$$

Enthalpy \(H\)

Enthalpy is a state function

$$H = E + PV$$

Under constant pressure

$$w = - P \Delta V$$
$$\Delta E = q_P - P\Delta E$$

Therefore, according to the def. of Enthalpy

$$q_P = \Delta E + P \Delta V = \Delta H$$

Calorimetry

Extensive means depends on the amount of substance. Intensive means not related. Heat Capacity \(C\) is an extensive property

Constant-Pressure Calorimetry

\(s\) means specific heat capacity (per g).

$$\Delta H = q_P = s \times m \times \Delta T$$

Constant-Volume Calorimetry

Because const. V, work is zero

$$\Delta E = q + w = q_V = \Delta T \times C$$

Entropy \(S\)

A measure of molecular randomness or disorder. Positional Probability and Positional Microstates Boltzmann Entropy Formula (\(W\) corresponds to the number of microstates)

$$S = k_b \ln W$$

Second Law of Thermodynamics

In any spontaneous process there is always an increase in the entropy of the universe.

$$\Delta S_{univ} = \Delta S_{sys} + \Delta S_{surr}$$
$$\Delta S_{surr} = -\frac{\Delta H}{T}$$

Gibbs Free Energy

$$G = H - TS$$

Under const. T

$$\Delta G = \Delta H - T \Delta S$$
$$-\frac{\Delta G}{T} = -\frac{\Delta H}{T} + \Delta S = \Delta S_{sys} + \Delta S_{surr} = \Delta S_{univ}$$

A process at const. T and P is spontaneous only if \(\Delta G\) is negative.

  • \(\Delta H -\) and \(\Delta S +\) Spontaneous at All T
  • \(\Delta H +\) and \(\Delta S +\) Spontaneous at High T
  • \(\Delta H -\) and \(\Delta S -\) Spontaneous at Low T (Exothermicity is dominant)
  • \(\Delta H +\) and \(\Delta S -\) Not Spontaneous

Third Law of Thermodynamics

Entropy of a perfect crystal at 0 K is zero. Therefore, standard entropy \(S^0\) is normally greater than zero. However there are some special cases (\(F^-\) and \(OH^-\)) Generally, the more complex the molecule, the higher the standard entropy value (e.g. rotational and vibrational motions).

Standard Free Energy Change

$$\Delta G^0 = \Delta H^0 - T \Delta S^0$$

Entropy is a state function

$$\Delta S^0_{reaction} = \sum n_p S^0_p - \sum n_r S^0_r$$

Standard free energy of formation(\(\Delta G_f^0\))

Dependance of Free Energy on Pressure

For an ideal gas, enthalpy is not pressure-dependent, but entropy does depend on pressure because of volume.

$$G = G^0 + RT \ln P$$

Pf.

$$dE = TdS - PdV$$
$$dH = dE + PdV + VdP$$
$$dG = dH - TdS - SdT$$
$$dG = TdS - PdV + PdV + VdP - TdS - SdT = VdP$$

(Under const. T \(dT = 0\))

$$\int_i^f dG = \int_i^f VdP = \int_i^f \frac{RT}{P} dP = RT \int_i^f \ln P$$
$$G_f - G_i = RT\ln\frac{P_f}{P_i}$$

When \(G_i = G^0\) and \(P_i = 1\) (atm)

$$G - G^0 = RT\ln P$$

Using

$$\Delta G = \sum n_p G_p - \sum n_r G_r$$

We can get the relationship between \(\Delta G\) of a reaction and the reaction quotient \(Q\)

$$\Delta G = \Delta G^0 + RT\ln Q$$

Standard Free Energy of Formation \(\Delta G_f^0\) is defined as the change in free energy for the formation of 1 mol of a substance from its constituent elements in standard state.

Free Energy and Equilibrium

Equilibrium point occurs at the lowest value of free energy available to the reaction system \(G_{products} + G_reactants\). \(G_{products}\) will decrease and \(G_{reactant}\) will increase as long as their sum is decreasing, until \(\Delta G = 0\) and \(\Delta G_{products} = \Delta G_{reactants}\). At equilibrium \(Q = K\)

$$\Delta G = 0 = \Delta G^0 + RT \ln K$$
$$\Delta G^0 = -RT\ln K$$

When all reactants and products are 1 atm, if \(\Delta G^0 > 0\) then \(K < 1\) else if \(\Delta G^0 < 0\) then \(K > 0\).

Temperature Dependence of \(K\)

$$\ln K = \frac{\Delta G^0}{-RT} = -\frac{\Delta H^0}{R}(\frac{1}{T}) + \frac{\Delta S^0}{R}$$

Van't Hoff Equation

$$\ln (\frac{K_1}{K_2}) = -\frac{\Delta H^0}{R}(\frac{1}{T_2} - \frac{1}{T_1})$$

Free Energy and Work

$$w_{max} = \Delta G$$

For a spontaneous process \(\Delta G\) represents the energy that is free to do useful work. For a non spontaneous process \(\Delta G\) is the minimum work that must be spent to make the rpocess occur. Reversible process means the universe remains the same after the cyclic process. Any real pathway wastes energy, so all real processes are irreversible.

References

S. Zumdal, S. Zumdahl and H. Hsu. 2015. General Chemistry. Cengage Learning Asia

蔡蘊明 基礎普通化學 NTU OCW