**Problem Statement:**Prove that \[\left( \frac{\partial}{\partial T} \left( \frac{G}{T} \right) \right)_P = -\frac{H}{T^2}.\]**Explanation:**This expression is a thermodynamic relation involving partial derivatives. It is commonly used in the context of Gibbs free energy ($G$), temperature ($T$), and enthalpy ($H$).**Details:**- The left-hand side of the equation, $\left( \frac{\partial}{\partial T} \left( \frac{G}{T} \right) \right)_P$, represents the partial derivative of $ \frac{G}{T} $ with respect to temperature $T$, while holding pressure $P$ constant.- The right-hand side, $-\frac{H}{T^2}$, indicates that this derivative is equal to the negative of the enthalpy divided by the square of the temperature.This relation is essential for understanding the temperature dependence of Gibbs free energy in various thermodynamic processes.

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G+), --# H Prove that ƏT

Chemistry

10th Edition

ISBN:9781305957404

Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste

Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste

Chapter1: Chemical Foundations

Section: Chapter Questions

Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...

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Question

**Problem Statement:**

Prove that

\[
\left( \frac{\partial}{\partial T} \left( \frac{G}{T} \right) \right)_P = -\frac{H}{T^2}.
\]

**Explanation:**

This expression is a thermodynamic relation involving partial derivatives. It is commonly used in the context of Gibbs free energy ($G$), temperature ($T$), and enthalpy ($H$).

**Details:**

- The left-hand side of the equation, $\left( \frac{\partial}{\partial T} \left( \frac{G}{T} \right) \right)_P$, represents the partial derivative of $ \frac{G}{T} $ with respect to temperature $T$, while holding pressure $P$ constant.
- The right-hand side, $-\frac{H}{T^2}$, indicates that this derivative is equal to the negative of the enthalpy divided by the square of the temperature.

This relation is essential for understanding the temperature dependence of Gibbs free energy in various thermodynamic processes.

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