Question

Derive the relation \(c_{p}=-T\left(\partial^{2} g / \partial T^{2}\right)_{p}\).

Short answer

The relation is \(c_p = -T \left(\frac{\partial^2 g}{\partial T^2}\right)_p\).

Step-by-step solution

Understand Gibbs Free Energy

Recall that the Gibbs free energy is defined as \( g = h - Ts \), where \(h\) is enthalpy and \(s\) is entropy.

Use Gibbs Free Energy as a Function of T and p

The differential form of Gibbs free energy in terms of temperature \(T\) and pressure \(p\) is given by \( dg = -s dT + v dp \), where \(v\) is volume.

Recognize the Second Derivative of g with respect to T

The specific heat at constant pressure, \( c_p \), can be related to Gibbs free energy by examining the second derivative of \( g \) with respect to \( T \) at constant pressure. Specifically, we focus on the entropy term, since \( s = -\left(\frac{\partial g}{\partial T}\right)_p \).

Apply the Thermodynamic Relation

The specific heat at constant pressure is given by \( c_p = T \left(\frac{\partial s}{\partial T}\right)_p \). Recall from Step 3 that \( s = -\left(\frac{\partial g}{\partial T}\right)_p \), so \( \left(\frac{\partial s}{\partial T}\right)_p = -\left(\frac{\partial^2 g}{\partial T^2}\right)_p \).

Derive the Final Relation

Substitute the expression for \( \left(\frac{\partial s}{\partial T}\right)_p \) into the equation for \( c_p \): \( c_p = T \left( - \left(\frac{\partial^2 g}{\partial T^2}\right)_p \right) \). Simplifying, we get the desired relation: \( c_p = -T \left( \frac{\partial^2 g}{\partial T^2} \right)_p \).

Key concepts

Gibbs Free Energy

Gibbs free energy, denoted as \( G \) or \( g \), is a vital concept in thermodynamics. It is defined as the thermodynamic potential that measures the maximum reversible work obtainable from a thermodynamic system at constant temperature and pressure. Formally, it's expressed as \( g = h - Ts \), where \( h \) is enthalpy, \( T \) is temperature, and \( s \) is entropy.
This equation highlights that Gibbs free energy accounts for the total energy and the energy dispersal within a system.
In practical applications, Gibbs free energy helps predict the direction of chemical reactions and phase changes.
Now, let’s recall that the differential form of Gibbs free energy for a system in terms of temperature and pressure is given by \( dg = -s dT + v dp \), where \( v \) is volume.

Specific Heat at Constant Pressure

Specific heat at constant pressure, denoted by \( c_p \), is the amount of heat needed to raise the temperature of a unit mass of a substance by one degree Celsius at constant pressure.
This property is essential in studying how substances respond to heating under constant pressure conditions typical in many practical scenarios, like atmospheric processes.
Mathematically, \( c_p = T \left(\frac{\partial s}{\partial T}\right)_p \).
Let's integrate this with our earlier information.
From Gibbs free energy, recall that \( s = -\left(\frac{\partial g}{\partial T}\right)_p \). Therefore, \( \left(\frac{\partial s}{\partial T}\right)_p = -\left(\frac{\partial^2 g}{\partial T^2}\right)_p \).
Substituting this into the equation for \( c_p \), we get \( c_p = T \left( - \left(\frac{\partial^2 g}{\partial T^2} \right)_p \right) \), which simplifies to \( c_p = -T \left( \frac{\partial^2 g}{\partial T^2} \right)_p \).

Thermodynamic Relations

Thermodynamic relations are equations that connect different thermodynamic properties.
These relations are derived from the fundamental thermodynamic potentials like internal energy \( U \), enthalpy \( H \), Helmholtz free energy \( F \), and Gibbs free energy \( G \). For instance:

• The Maxwell relations, which stem from the second derivatives of these thermodynamic potentials.
• The relationships between heat capacities \( c_p \) and \( c_v \) with the thermodynamic properties.

In the context of Gibbs free energy, one useful relation we've dealt with is the expression of entropy as \( s = - \left(\frac{\partial g}{\partial T}\right)_p \). This can be further differentiated to link specific heat at constant pressure and the derivatives of \( g \).

Partial Derivatives in Thermodynamics

Partial derivatives play a crucial role in thermodynamics. They help describe how a system's properties change in response to variations in other variables.
For example, the partial derivative \( \left(\frac{\partial g}{\partial T}\right)_p \) tells how Gibbs free energy changes with temperature at constant pressure.
In our specific case, we look at the second partial derivative of Gibbs free energy with respect to temperature:
\( \left( \frac{\partial^2 g}{\partial T^2} \right)_p \). This derivative represents how entropy changes with temperature, which is fundamental to understand the temperature dependence of specific heat at constant pressure.

Entropy and Enthalpy

Entropy (\( s \)) and enthalpy (\( h \)) are central concepts in thermodynamics.
Entropy represents the degree of disorder or randomness in a system, whereas enthalpy signifies the total heat content of a system.
Together, they form the basis for calculating Gibbs free energy: \( g = h - Ts \).
Enthalpy is particularly important in processes occurring at constant pressure, such as atmospheric processes and chemical reactions.
Entropy changes help us understand how energy is dispersed in a system, guiding us in predicting spontaneous processes.
In the derivation provided, understanding the roles of entropy and enthalpy in Gibbs free energy was crucial for connecting specific heat capacity to the second derivative of Gibbs free energy.