To determine the $\Delta G^\circ$ for the conversion of glucose ($C_6H_{12}O_6$) into lactic acid ($2 \, CH_3CH(OH)COOH$), we start by analyzing the given information and the reactions.

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Given:

1. $\Delta G^\circ$ for the phosphorylation of 2 moles of ADP to ATP during the glucose conversion is $-135 \, \mathrm{kJ/mol}$: $C_6H_{12}O_6(\text{aq}) + 2 \, HPO_4^{2-}(\text{aq}) + 2 \, ADP^{3-}(\text{aq}) + 2 \, H^+(\text{aq}) \rightarrow 2 \, CH_3CH(OH)COOH(\text{aq}) + 2 \, ATP^{4-}(\text{aq}) + 2 \, H_2O(\text{l})$

2. $\Delta G^\circ$ for ATP hydrolysis is $-30.5 \, \mathrm{kJ/mol}$: $ATP^{4-}(\text{aq}) + H_2O(\text{l}) \rightarrow ADP^{3-}(\text{aq}) + HPO_4^{2-}(\text{aq}) + H^+(\text{aq})$

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Step 1: Energy contribution of ATP synthesis

In the glucose-to-lactic acid reaction, 2 moles of ATP are synthesized. Since the hydrolysis of 1 mole of ATP releases $-30.5 \, \mathrm{kJ/mol}$, synthesizing 1 mole of ATP requires $+30.5 \, \mathrm{kJ/mol}$. For 2 moles: $\Delta G^\circ \text{ for ATP synthesis} = 2 \times (+30.5) = +61.0 \, \mathrm{kJ/mol}.$

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Step 2: Determine $\Delta G^\circ$ for glucose-to-lactic acid conversion

The net energy released by the conversion of glucose into lactic acid, after accounting for ATP synthesis, is given as $-135 \, \mathrm{kJ/mol}$. Since this value includes the energy cost of ATP synthesis: $\Delta G^\circ (\text{glucose to lactic acid only}) = \Delta G^\circ (\text{overall}) + \Delta G^\circ (\text{ATP synthesis}).$

Substitute the values: $\Delta G^\circ (\text{glucose to lactic acid only}) = -135 + 61.0 = -196 \, \mathrm{kJ/mol}.$

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Final Answer:

The $\Delta G^\circ$ for the conversion of glucose into lactic acid ($C_6H_{12}O_6 \rightarrow 2 \, CH_3CH(OH)COOH$) is: $\boxed{-196 \, \mathrm{kJ/mol}}.$

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Let me know if you'd like further clarifications or have questions! 😊

Related Questions:

1. How does the energy cost of ATP synthesis compare to ATP hydrolysis?
2. Why is $\Delta G^\circ$ negative for exergonic reactions like glucose breakdown?
3. What role does phosphorylation play in cellular metabolism?
4. How can the efficiency of ATP production from glucose be quantified?
5. How would $\Delta G^\circ$ change if fewer moles of ATP were synthesized?

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Tip: When solving thermodynamic problems, carefully isolate each energy component for clarity!