Gibbs Free energy calculations - Free Printable

Let’s solve each problem step by step. We’ll use the Gibbs free energy formula:

ΔG = ΔH – TΔS

Where:
- ΔG = Gibbs free energy change (what we’re solving for)
- ΔH = change in heat (enthalpy), given in cal
- T = temperature in Kelvin (K) — we must convert from °F to K
- ΔS = change in entropy, given in cal/K

We are also given conversion formulas:
- °C = (°F – 32) × 5/9
- K = °C + 273

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Problem 1: ATP → ADP + Pi

Given:
- Temperature = 68 °F
- ΔH = 19,070 cal
- ΔS = 90 cal/K

Step 1: Convert 68 °F to Kelvin

First, convert to Celsius:
°C = (68 – 32) × 5/9 = 36 × 5/9 = 20 °C

Now convert to Kelvin:
K = 20 + 273 = 293 K

Step 2: Plug into ΔG = ΔH – TΔS

ΔG = 19,070 – (293 × 90)

Calculate TΔS:
293 × 90 = 26,370

Now subtract:
ΔG = 19,070 – 26,370 = –7,300 cal

✔ So, ΔG = –7,300 cal

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Problem 2: glutamate + NH₃ → glutamine + H₂O

Given:
- Temperature = 68 °F → same as above → 293 K
- ΔH = 4,103 cal
- ΔS = 2.4 cal/K

Step 1: Use same T = 293 K

Step 2: Plug into ΔG = ΔH – TΔS

ΔG = 4,103 – (293 × 2.4)

Calculate TΔS:
293 × 2.4 = let’s compute:

293 × 2 = 586
293 × 0.4 = 117.2
Total = 586 + 117.2 = 703.2

Now subtract:
ΔG = 4,103 – 703.2 = 3,399.8 cal ≈ 3,400 cal (rounded)

✔ So, ΔG = +3,400 cal

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Problem 3: Would either reaction occur spontaneously?

A reaction is spontaneous if ΔG < 0 (negative).

- Reaction 1: ΔG = –7,300 cal → spontaneous
- Reaction 2: ΔG = +3,400 cal → not spontaneous

✔ Answer: Only the first reaction (ATP → ADP + Pi) occurs spontaneously because its ΔG is negative.

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Problem 4: Are either of the reactions endergonic?

An endergonic reaction has ΔG > 0 (positive) — it requires energy input.

- Reaction 1: ΔG = –7,300 → exergonic
- Reaction 2: ΔG = +3,400 → endergonic

✔ Answer: Yes, the second reaction (glutamate + NH₃ → glutamine + H₂O) is endergonic because its ΔG is positive.

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Problem 5: How does ΔG change if temperature rises to 98.6 °F?

Convert 98.6 °F to Kelvin:

°C = (98.6 – 32) × 5/9 = 66.6 × 5/9 = 37 °C
K = 37 + 273 = 310 K

Now recalculate ΔG for both reactions at T = 310 K.

#### Reaction 1: ATP → ADP + Pi

ΔG = ΔH – TΔS = 19,070 – (310 × 90)

310 × 90 = 27,900

ΔG = 19,070 – 27,900 = –8,830 cal

Originally at 293 K: –7,300 cal
Now at 310 K: –8,830 cal → more negative

→ So, ΔG becomes more negative → reaction becomes *more* spontaneous.

#### Reaction 2: glutamate + NH₃ → glutamine + H₂O

ΔG = 4,103 – (310 × 2.4)

310 × 2.4 = 744

ΔG = 4,103 – 744 = 3,359 cal

Originally at 293 K: ~3,400 cal
Now at 310 K: ~3,359 cal → slightly less positive

→ Still positive, but smaller → still not spontaneous, but closer to being spontaneous.

✔ Answer: For reaction 1, ΔG becomes more negative (–8,830 cal). For reaction 2, ΔG becomes slightly less positive (~3,359 cal). Both changes make the reactions “more favorable” thermodynamically, though only reaction 1 remains spontaneous.

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Problem 6: Does increasing temperature make each reaction more or less likely to occur spontaneously?

Recall: Spontaneity depends on sign of ΔG.

The formula is ΔG = ΔH – TΔS

So, how T affects ΔG depends on the sign of ΔS.

#### Reaction 1: ΔS = +90 cal/K (positive)

As T increases, TΔS increases → so we subtract a larger number → ΔG becomes more negative → more spontaneous

#### Reaction 2: ΔS = +2.4 cal/K (also positive!)

Same logic: As T increases, TΔS increases → we subtract a larger number → ΔG decreases (becomes less positive) → closer to spontaneous, but still not spontaneous unless T gets very high.

Wait — let’s check: Could reaction 2 ever become spontaneous?

Set ΔG = 0:

0 = 4,103 – T×2.4 → T = 4,103 / 2.4 ≈ 1,709 K → way too hot! Not biologically relevant.

But the question is just about whether increasing T makes them *more or less likely* to be spontaneous.

Since both have positive ΔS, increasing T makes ΔG decrease (more negative or less positive) → so both become more likely to be spontaneous as T increases.

✔ Answer: Increasing temperature makes both reactions more likely to occur spontaneously because both have positive entropy changes (ΔS > 0), so higher T reduces ΔG.

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Final Answer:

1. ΔG = –7,300 cal
2. ΔG = +3,400 cal
3. Only reaction 1 (ATP → ADP + Pi) is spontaneous because ΔG < 0.
4. Only reaction 2 (glutamate + NH₃ → glutamine + H₂O) is endergonic because ΔG > 0.
5. At 98.6°F (310 K):
- Reaction 1: ΔG = –8,830 cal (more negative)
- Reaction 2: ΔG ≈ 3,359 cal (slightly less positive)
6. Increasing temperature makes both reactions more likely to be spontaneous because both have positive ΔS, so higher T lowers ΔG.