To tackle your questions, let's break them down step by step, starting with the equilibrium constant for the hydrolysis of sucrose and then moving on to the reaction order based on the rate changes you described.
Understanding the Equilibrium Constant
The equilibrium constant, denoted as Kc, is a crucial concept in chemical thermodynamics. For the hydrolysis of sucrose, which can be represented as:
- C12H22O11 + H2O ⇌ C6H12O6 + C6H12O6
Given that Kc is 2 x 10^23 at 300 K, this indicates that at equilibrium, the concentration of products (glucose and fructose) is significantly higher than that of the reactants (sucrose and water). A Kc value this large suggests that the reaction strongly favors the formation of products, meaning that the hydrolysis of sucrose is highly spontaneous under these conditions.
Calculating Gibbs Free Energy Change
The Gibbs free energy change (ΔG) for a reaction can be calculated using the equation:
ΔG = -RT ln(Kc)
Where:
- R = 8.314 J/(mol·K) (the universal gas constant)
- T = temperature in Kelvin (300 K in this case)
- ln(Kc) = natural logarithm of the equilibrium constant
Plugging in the values:
ΔG = - (8.314 J/(mol·K) * 300 K) * ln(2 x 10^23)
Calculating ln(2 x 10^23) gives approximately 52.3. Therefore:
ΔG = - (8.314 * 300) * 52.3 ≈ -130,000 J/mol
This negative value indicates that the reaction is spontaneous under standard conditions at 300 K.
Determining the Overall Order of the Reaction
Now, let's analyze the reaction aG + bH → products based on the rate changes you provided. The information states:
- When the concentrations of both G and H are doubled, the rate increases 8 times.
- When the concentration of G is doubled while keeping H fixed, the rate doubles.
From the first observation, we can infer that the rate law can be expressed as:
Rate = k[G]^m[H]^n
When both concentrations are doubled:
Rate' = k(2G)^m(2H)^n = 2^m * 2^n * k[G]^m[H]^n = 2^(m+n) * Rate
Given that the rate increases 8 times, we have:
2^(m+n) = 8
This implies that:
m + n = 3
For the second observation, when only G is doubled:
Rate' = k(2G)^m[H]^n = 2^m * k[G]^m[H]^n = 2^m * Rate
Since the rate doubles, we have:
2^m = 2
This means:
m = 1
Now, substituting m back into the first equation:
1 + n = 3
Thus, we find:
n = 2
Final Reaction Order
The overall order of the reaction is the sum of the individual orders with respect to each reactant:
Overall order = m + n = 1 + 2 = 3
In summary, the overall order of the reaction is 3, indicating a third-order reaction where the rate depends on the concentration of G to the first power and H to the second power. This means that the reaction rate is quite sensitive to changes in the concentrations of the reactants.
