Thermodynamics relation — Interpreting ΔG° = -RTln(K): effect of a tenfold change in the equilibrium constant
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Aa 10-fold increase in K decreases ΔG° by about 10-fold
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Ba 10-fold decrease in K decreases ΔG° by about 2.3RT
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Ca 10-fold increase in K decreases ΔG° by about 2.3RT
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Da 10-fold decrease in K increases ΔG° by about 10-fold
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Ea 2-fold increase in K decreases ΔG° by about RT
Answer
Correct Answer: a 10-fold increase in K decreases ΔG° by about 2.3RT
Explanation
Introduction:The standard free energy change links directly to the equilibrium constant through the fundamental relation ΔG° = -RTln(K). This question checks quick, order of magnitude reasoning for decimal changes in K without a calculator.
Given Data / Assumptions:
- R is the gas constant and T is absolute temperature.
- Natural logarithm is used in the equation.
- Goal: estimate the change in ΔG° when K changes by a factor of 10.
Concept / Approach:Use the identity ln(10) ≈ 2.303. Substituting into ΔG° = -RTln(K) shows how a tenfold change in K maps to a fixed multiple of RT independent of the actual K value, aside from sign direction.
Step-by-Step Solution:
1) Start with ΔG° = -RTln(K).2) For a 10-fold increase, K becomes 10K_old, so ln(K_new) = ln(10) + ln(K_old).3) The change in ΔG° is -RTln(10) ≈ -2.3RT.4) Therefore, ΔG° decreases by about 2.3RT for a 10-fold increase in K.Verification / Alternative check:If K decreases by 10, then ln(K) decreases by 2.3, so ΔG° increases by about 2.3RT. The direction is consistent with favorable equilibria (larger K) corresponding to more negative ΔG°.
Why Other Options Are Wrong:
- 10-fold change in ΔG°: free energy is not proportional to K linearly.
- Decrease in K decreases ΔG°: sign is wrong; ΔG° becomes less negative.
- 10-fold decrease increases ΔG° by 10-fold: again mixes linear and logarithmic relations.
- 2-fold increase gives RT: ln(2) is about 0.693, not 1.0, so the magnitude is off.
Common Pitfalls:Mixing log base 10 and natural log. Always convert decimal folds to natural logs with ln(10) ≈ 2.303 for quick estimates.
Final Answer:a 10-fold increase in K decreases ΔG° by about 2.3RT.