What is the mathematical form of Gibbs Helmholtz equation?
2 Answers
The Gibbs-Helmholtz equation relates the Gibbs free energy \( G \) of a system to its enthalpy \( H \) and entropy \( S \). It provides insights into how the free energy changes with temperature. The equation is derived from the fundamental thermodynamic relationship between these quantities.
The mathematical form of the Gibbs-Helmholtz equation is:
\[ \left( \frac{\partial G}{\partial T} \right)_P = -S \]
where:
- \( G \) is the Gibbs free energy,
- \( T \) is the absolute temperature,
- \( P \) denotes constant pressure,
- \( S \) is the entropy of the system.
This equation indicates that the temperature dependence of the Gibbs free energy is directly related to the entropy of the system. To understand this in context:
1. **Gibbs Free Energy (\( G \))**: This is a thermodynamic potential that measures the maximum reversible work that can be performed by a system at constant temperature and pressure. It's defined as:
\[ G = H - TS \]
where \( H \) is the enthalpy and \( T \) is the absolute temperature.
2. **Enthalpy (\( H \))**: This is the total heat content of the system and is defined as:
\[ H = U + PV \]
where \( U \) is the internal energy, \( P \) is the pressure, and \( V \) is the volume.
3. **Entropy (\( S \))**: This measures the disorder or randomness in the system and reflects the number of ways the system can be arranged without changing its energy.
The Gibbs-Helmholtz equation can be integrated to yield another useful form when considering changes in temperature:
\[ \frac{\Delta G}{T} = -\Delta S + \text{constant} \]
where:
- \( \Delta G \) is the change in Gibbs free energy,
- \( \Delta S \) is the change in entropy.
This form shows that the change in Gibbs free energy with respect to temperature involves the change in entropy, providing a direct link between these thermodynamic properties.
The mathematical form of the Gibbs-Helmholtz equation is:
\[ \left( \frac{\partial G}{\partial T} \right)_P = -S \]
where:
- \( G \) is the Gibbs free energy,
- \( T \) is the absolute temperature,
- \( P \) denotes constant pressure,
- \( S \) is the entropy of the system.
This equation indicates that the temperature dependence of the Gibbs free energy is directly related to the entropy of the system. To understand this in context:
1. **Gibbs Free Energy (\( G \))**: This is a thermodynamic potential that measures the maximum reversible work that can be performed by a system at constant temperature and pressure. It's defined as:
\[ G = H - TS \]
where \( H \) is the enthalpy and \( T \) is the absolute temperature.
2. **Enthalpy (\( H \))**: This is the total heat content of the system and is defined as:
\[ H = U + PV \]
where \( U \) is the internal energy, \( P \) is the pressure, and \( V \) is the volume.
3. **Entropy (\( S \))**: This measures the disorder or randomness in the system and reflects the number of ways the system can be arranged without changing its energy.
The Gibbs-Helmholtz equation can be integrated to yield another useful form when considering changes in temperature:
\[ \frac{\Delta G}{T} = -\Delta S + \text{constant} \]
where:
- \( \Delta G \) is the change in Gibbs free energy,
- \( \Delta S \) is the change in entropy.
This form shows that the change in Gibbs free energy with respect to temperature involves the change in entropy, providing a direct link between these thermodynamic properties.
The Gibbs-Helmholtz equation is a fundamental relation in thermodynamics that connects the change in the Gibbs free energy of a system to the change in its enthalpy and entropy. It’s particularly useful for understanding how the free energy of a system changes with temperature.
Here's the mathematical form of the Gibbs-Helmholtz equation:
\[ \left( \frac{\partial \left( \frac{G}{T} \right)}{\partial T} \right)_P = -\frac{\Delta H}{T^2} \]
where:
- \( G \) is the Gibbs free energy.
- \( T \) is the absolute temperature.
- \( \Delta H \) is the enthalpy change of the system.
**Explanation of Terms:**
- **Gibbs Free Energy (G):** This is a thermodynamic potential that measures the maximum reversible work that can be performed by a system at constant temperature and pressure.
- **Temperature (T):** It is the absolute temperature of the system, measured in Kelvin.
- **Enthalpy Change (ΔH):** This is the heat content of the system, indicating the amount of heat absorbed or released during a process.
**Derivation and Application:**
1. **Starting Point:**
The Gibbs free energy \( G \) is defined as:
\[ G = H - TS \]
where:
- \( H \) is the enthalpy.
- \( S \) is the entropy.
2. **Differentiate \( G \):**
To find how \( G \) changes with temperature, differentiate \( G \) with respect to \( T \):
\[ \frac{\partial G}{\partial T} = \frac{\partial (H - TS)}{\partial T} \]
Applying the product rule for differentiation:
\[ \frac{\partial G}{\partial T} = \frac{\partial H}{\partial T} - \frac{\partial (TS)}{\partial T} \]
Since \( H \) can be a function of temperature, and assuming that \( \frac{\partial H}{\partial T} \) is \( \Delta H \) (enthalpy change):
\[ \frac{\partial G}{\partial T} = \Delta H - \left( S + T \frac{\partial S}{\partial T} \right) \]
\[ \frac{\partial G}{\partial T} = \Delta H - S \]
3. **Free Energy per Unit Temperature:**
Express \( \frac{G}{T} \) and its derivative with respect to \( T \):
\[ \frac{G}{T} = \frac{H - TS}{T} = \frac{H}{T} - S \]
Differentiate with respect to \( T \):
\[ \frac{\partial}{\partial T} \left( \frac{G}{T} \right) = \frac{\partial}{\partial T} \left( \frac{H}{T} - S \right) \]
\[ \frac{\partial}{\partial T} \left( \frac{H}{T} - S \right) = \frac{\partial \left( \frac{H}{T} \right)}{\partial T} - \frac{\partial S}{\partial T} \]
\[ \frac{\partial \left( \frac{H}{T} \right)}{\partial T} = \frac{1}{T} \frac{\partial H}{\partial T} - \frac{H}{T^2} \]
\[ \frac{\partial \left( \frac{G}{T} \right)}{\partial T} = \frac{1}{T} \frac{\partial H}{\partial T} - \frac{H}{T^2} - \frac{\partial S}{\partial T} \]
Given \( \frac{\partial G}{\partial T} = \Delta H - S \):
\[ \frac{\partial \left( \frac{G}{T} \right)}{\partial T} = -\frac{\Delta H}{T^2} \]
This final form of the Gibbs-Helmholtz equation shows how the Gibbs free energy \( G \) changes with temperature, providing insight into the temperature dependence of enthalpy and entropy in a system.
Here's the mathematical form of the Gibbs-Helmholtz equation:
\[ \left( \frac{\partial \left( \frac{G}{T} \right)}{\partial T} \right)_P = -\frac{\Delta H}{T^2} \]
where:
- \( G \) is the Gibbs free energy.
- \( T \) is the absolute temperature.
- \( \Delta H \) is the enthalpy change of the system.
**Explanation of Terms:**
- **Gibbs Free Energy (G):** This is a thermodynamic potential that measures the maximum reversible work that can be performed by a system at constant temperature and pressure.
- **Temperature (T):** It is the absolute temperature of the system, measured in Kelvin.
- **Enthalpy Change (ΔH):** This is the heat content of the system, indicating the amount of heat absorbed or released during a process.
**Derivation and Application:**
1. **Starting Point:**
The Gibbs free energy \( G \) is defined as:
\[ G = H - TS \]
where:
- \( H \) is the enthalpy.
- \( S \) is the entropy.
2. **Differentiate \( G \):**
To find how \( G \) changes with temperature, differentiate \( G \) with respect to \( T \):
\[ \frac{\partial G}{\partial T} = \frac{\partial (H - TS)}{\partial T} \]
Applying the product rule for differentiation:
\[ \frac{\partial G}{\partial T} = \frac{\partial H}{\partial T} - \frac{\partial (TS)}{\partial T} \]
Since \( H \) can be a function of temperature, and assuming that \( \frac{\partial H}{\partial T} \) is \( \Delta H \) (enthalpy change):
\[ \frac{\partial G}{\partial T} = \Delta H - \left( S + T \frac{\partial S}{\partial T} \right) \]
\[ \frac{\partial G}{\partial T} = \Delta H - S \]
3. **Free Energy per Unit Temperature:**
Express \( \frac{G}{T} \) and its derivative with respect to \( T \):
\[ \frac{G}{T} = \frac{H - TS}{T} = \frac{H}{T} - S \]
Differentiate with respect to \( T \):
\[ \frac{\partial}{\partial T} \left( \frac{G}{T} \right) = \frac{\partial}{\partial T} \left( \frac{H}{T} - S \right) \]
\[ \frac{\partial}{\partial T} \left( \frac{H}{T} - S \right) = \frac{\partial \left( \frac{H}{T} \right)}{\partial T} - \frac{\partial S}{\partial T} \]
\[ \frac{\partial \left( \frac{H}{T} \right)}{\partial T} = \frac{1}{T} \frac{\partial H}{\partial T} - \frac{H}{T^2} \]
\[ \frac{\partial \left( \frac{G}{T} \right)}{\partial T} = \frac{1}{T} \frac{\partial H}{\partial T} - \frac{H}{T^2} - \frac{\partial S}{\partial T} \]
Given \( \frac{\partial G}{\partial T} = \Delta H - S \):
\[ \frac{\partial \left( \frac{G}{T} \right)}{\partial T} = -\frac{\Delta H}{T^2} \]
This final form of the Gibbs-Helmholtz equation shows how the Gibbs free energy \( G \) changes with temperature, providing insight into the temperature dependence of enthalpy and entropy in a system.