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Since the changes of entropy of chemical reaction are not measured readily, thus, entropy is not used as a criterion. To obviate this difficulty, we use a new thermodynamic function called the free energy whose symbol is G.
In 18th century, Josiah Willard Gibbs, one of the founders of thermodynamics, combined the first law with the second law of thermodynamics and created the free energy function of a system, which has the following equation:
\[ \Delta{G} = \Delta H - T \Delta S \tag{1}\]
\( \Delta G \): The change in free energy undergoes a transformation at constant temperature and constant pressure.
\( \Delta H \): The change in enthalpy
\( \Delta S \): The change in entropy
If \( \left | \Delta H \right | >> \left | T\Delta S \right |\): the reaction is enthalpy-driven
If \( \Delta H \) << \( T\Delta S \): the reaction is entropy-driven
Note: constant pressure q = \( \Delta H \) and constant temperature are required to derive equation (1).
The enthalpy change is \( \Delta H = \Delta E -P \Delta V \).
For most biochemical reactions, \( \Delta V \) (the volume change), is considered very small so that we can ignore it. Thus, we have \( \Delta H \) is equal to \( \Delta U \), which leads to:
\[ \Delta G \cong \Delta U - T\Delta S \]
Hence, there are two factors affect the change in free energy \( \Delta G \):
These factors are contrast to each other, which is a important criterion to determine a reaction is spontaneous or not.
The affactors affect \( \Delta G \)of a reaction (assume \( \Delta H \) and \( \Delta S \) are independent of temperature):
\(\Delta H\) | \(\Delta S\) | \(\Delta G\) | Example |
---|---|---|---|
+ | + | at low temperature: + , at high temperature: - | 2HgO(s) -> 2Hg (l) + O_{2} (g) |
+ | - | at all temperature: + | 3O_{2} (g) ->2O_{3} (g) |
- | + | at all temperature: - | 2H_{2}O_{2} (l) -> 2H_{2}O (l) + O_{2} (g) |
- | - | at low temperature: - , at high temperature: + | NH_{3} (g) + HCl (g) -> NH_{4}Cl (s) |
Note:
The standard Gibbs energy change \( \Delta G^o \) (at which reactants are converted to products at 1 bar) for:
\[ aA + bB \rightarrow cC + dD \]
\[ \Delta r G^o = c \Delta _fG^o (C) + d \Delta _fG^o (D) - a \Delta _fG^o (A) - b \Delta _fG^o (B) \]
\[\Delta _fG^0 = \sum v \Delta _f G^0 (\text {products}) - \sum v \Delta _f G^0 (\text {reactants}) \]
Let's consider the following reaction:
\[ A + B \leftrightharpoons C + D\]
We will obtain \(\Delta{G}\):
\[\Delta{G} = \Delta{G}^o + RT\ ln \dfrac{[C][D]}{[A][B]} \tag{2}\]
with
The Gibb's free energy \(\Delta{G}\) depends primarily on the reactants' nature and concentrations (expressed in the \(\Delta{G}^o\) term and the logarithmic term of equation (2), respectively).
At the standard state, pH = 7, the standard free change is denoted as \(\Delta{G}^{o'}\).
At equilibrium, \(\Delta{G} = 0\): no driving force remains
\[0 = \Delta{G}^{o'} + RTln\dfrac{[C][D]}{[A][B]}\]
\[\Delta{G}^{o'} = -RTln\dfrac{[C][D]}{[A][B]}\] (3)
The equilibrium constant, \(K'eq = \dfrac{[C][D]}{[A][B]}\)
When \(K'eq\) is large, almost all reactants are converted to products.
Substituting \[K'eq\] into equation (3), we have:
\[\Delta{G}^{o'} = -RTlnK'eq\]
or
\[\Delta{G}^{o'} = -2.303RT log_{10} K'eq\]
Rearrange,
\[K'eq = 10^{-\Delta{G}^{o'}/(2.303RT)}\]
Let's calculate \[\Delta{G}^{o'}\]. For example, what is \(\Delta{G}^{o'}\) for isomerization of dihydroxyacetone phosphate to glyceraldehyde 3-phosphate? At equilibrium, we have \[K'eq\] = 0.0475 at 298 K and pH 7. We can calculate:
\[\Delta{G}^{o'} = -2.303RT log_{10} K'eq\] = (-2.303) * (1.98 * 10^{-3}) * 298 * (log_{10}0.0475) = 1.8 kcal/mol
Given:
From equation (2),
\(\Delta{G}\) = 1.8 kcal/mol + 2.303 RT log_{10}(3*10^{-6} M/2*10^{-4} M) = -0.7 kcal/mol
Note: \(\Delta{G} < 0 \): the reaction occurs spontaneously when \(\Delta{G}^{o'} > 0\). A reaction is determined spontaneous or not depends on \(\Delta{G}\), not \(\Delta{G}^{o'}\).
\[S (Substrate) \leftrightharpoons S^{\ddagger} (Transition State) \rightarrow P (Product)\]
The transition state has a higher free energy than either substrate or product.
The Gibbs free energy of activation is denoted as \(E(A\) or \(\Delta{G}^{\ddagger}\) is equal to transition state's free energy minus substrate's free energy. In catalyzed reaction, enzyme accelerate reaction by decreasing the Gibbs free energy of activation (the activation barrier) or by stabilizing transition states.
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