Problem 15 - A Modern Course in Statistical Physics

Question

The entropy production, \(\sigma\), in a thermocouple can be written $$ T \sigma=-J_{S} \cdot \nabla_{r} T-I \cdot\left(\nabla_{r} \frac{\mu_{\text {el }}}{\boldsymbol{F}}-\boldsymbol{F} \boldsymbol{E}\right) $$ where \(J_{S}\) is the entropy current, \(I\) is the current carried by the electrons, \(F\) is Faraday's constant, \(E\) is the electric field in the metal wires, and \(\mu_{\mathrm{el}}\) is the chemical potential of the electrons. The balance equation for the entropy/volume, \(s\), is \(\partial s / \partial t=-\nabla \cdot J_{S}+\sigma\). The generalized Ohm's laws can be written $$ \boldsymbol{J}_{S}=-\frac{\lambda}{T} \nabla, T+\frac{\Gamma}{T} I \text { and } \quad E-\nabla_{r} \frac{\mu_{e l}}{F}=-\zeta \Gamma_{r} T+R I $$ where \(\lambda\) is the coefficient of thermal conductivity at zero electrical current, \(R\) is the isothermal electrical resistance, \(\zeta\) is the differential thermoelectric power, and \(\Gamma / T\) is the entropy transported per unit electric current. (a) Show that the Onsager relation, \(L_{S E}=L_{E S}\), implies that \(\Gamma=T \zeta\), (b) Show that the entropy balance equation can be written \(\partial s / \partial t=\lambda / T \nabla_{r}^{2}(T)-\nabla_{r} \cdot \Gamma I / T+R I^{2} / T\), The first term on the right is the entropy production due to thermal conductivity. The third term on the right is the entropy production due to Joule heating. The second term on the right is the entropy production due to the Peltier and Thomson effects.

Step-by-Step Solution

Answer

a) \Gamma = T \zeta\. b) \frac{\partial s}{\partial t} = \frac{\lambda}{T} abla_{r}^{2}(T) - abla_{r} \cdot \left(\frac{\Gamma}{T} I \right) + \frac{R I^{2}}{T}.\

Step 1: Express Onsager Relation

The Onsager relation states that the cross-coefficients in a thermodynamical system are equal, i.e., \(L_{SE} = L_{ES}\). Based on the generalized Ohm's laws, identify \(L_{SE} \) and \(L_{ES}\).

Step 2: Substitute Onsager Coefficients

From the generalized Ohm's laws, we observe that: \[\boldsymbol{J}_{S} = -\frac{\lambda}{T} abla T + \frac{\Gamma}{T} I\] and \[E - abla_{r} \frac{\mu_{el}}{F} = -\zeta \Gamma_{r} T + R I\]. Substituting these, we identify \(L_{SE} = \frac{\Gamma}{T}\) and \(L_{ES} = \zeta \).

Step 3: Equate the Coefficients

Since \(L_{SE} = L_{ES}\), equate the coefficients: \[\frac{\Gamma}{T} = \zeta\]. Therefore, \Gamma = T \zeta\.

Step 4: Start with the Entropy Balance Equation

Given the entropy balance equation: \[\frac{\partial s}{\partial t} = - abla \cdot J_{S} + \sigma\], substitute the expression for \(J_{S} \) from the generalized Ohm's law.

Step 5: Substitute \(J_{S}\) in the Entropy Balance Equation

Using \[\boldsymbol{J}_{S} = -\frac{\lambda}{T} abla T + \frac{\Gamma}{T} I\], substitute into the entropy balance equation to get: \[- abla \cdot (-\frac{\lambda}{T} abla T + \frac{\Gamma}{T} I) + \sigma\].

Step 6: Simplify the Equation

Simplify the expression: \[\frac{\partial s}{\partial t} = abla \cdot \left(\frac{\lambda}{T} abla T \right) - abla \cdot \left(\frac{\Gamma}{T} I \right) + \sigma.\]

Step 7: Substitute for \sigma\

Substitute \( \sigma \) from the original equation: \[T \sigma = -J_{S} \cdot abla_{r} T - I \cdot \left(abla_{r} \frac{\mu_{el}}{F} - \mathbf{E} \right).\]

Step 8: Substitute Generalized Ohm’s Law

Using the generalized Ohm's law: \[E - abla_{r} \frac{\mu_{el}}{F} = -\zeta \Gamma_{r} T + R I,\] substitute \(\sigma \) back into the balance equation.

Step 9: Final Form of the Equation

Combine and simplify the expressions to get: \[\frac{\partial s}{\partial t} = \frac{\lambda}{T} abla_{r}^{2}(T) - abla_{r} \cdot \left(\frac{\Gamma}{T} I \right) + \frac{R I^{2}}{T}.\]

Key Concepts

entropy production

Entropy production is a key concept in thermodynamics, representing the increase in entropy within a system due to irreversible processes. It symbolizes the 'disorder' introduced into a system when energy is transferred or transformed in a non-reversible manner. For a thermocouple, the entropy production, \(\sigma\), can be expressed as: \[ T \sigma=-J_{S} \cdot \abla_{r} T-I \cdot\left(\abla_{r} \frac{\mu_{\text {el }}}{\boldsymbol{F}}-\boldsymbol{F} \boldsymbol{E}\right) \] Here, each term contributes to the total entropy production:

• \( J_{S} \) is the entropy current.
• \( \abla_{r} T \) represents the spatial temperature variation.
• \( I \) is the electric current.
• \( \mu_{\text {el}} \) is the electron chemical potential.
• \( F \) is Faraday's constant.
• \( E \) stands for the electric field.
  The entropy production due to these variables is critical in analyzing how systems evolve over time, especially in irreversible processes. Understanding its components helps in formulating equations and laws governing energy and matter transfers.

Onsager relations

The Onsager relations are foundational in non-equilibrium thermodynamics. They express the linear relationship between fluxes (like heat or particle flux) and forces (like gradients of temperature or chemical potential). Essential in understanding transport phenomena, the Onsager relations state that the cross-coefficients are symmetric. For example, if \( L_{SE} \) is a coefficient describing entropy flow driven by an electric field, and \( L_{ES} \) is a coefficient describing an electric current driven by a temperature gradient, then Onsager's reciprocal relations state that \[ L_{SE} = L_{ES} \] Applying this to the generalized Ohm's laws:

• \( \boldsymbol{J}_{S} = -\frac{\lambda}{T} \abla T + \frac{\Gamma}{T} I \)
• \( E - \abla_{r} \frac{\mu_{el}}{F} = -\zeta \Gamma_{r} T + R I \)
  We identify:
  • \( L_{SE} = \frac{\Gamma}{T} \)
  • \( L_{ES} = \zeta \)
    Thus equivalence of these coefficients, as per Onsager's principle, implies \[ \Gamma = T \zeta \] indicating a direct link between thermoelectric effects and temperature gradients.

generalized Ohm's law

Generalized Ohm's law extends the conventional Ohm's law to encompass additional thermodynamic effects. It characterizes the behavior of electric current and entropy flows under thermal gradients and electric fields. In the context of thermoelectric materials, this law takes a broader form: \[ \boldsymbol{J}_{S} = -\frac{\lambda}{T} \abla T + \frac{\Gamma}{T} I \] represents the entropy current (\( J_S \)) driven by both temperature gradients and electric currents. \[ E - \abla_{r} \frac{\mu_{el}}{F} = -\zeta \Gamma_{r} T + R I \] describes the electric field (\( E \)) relative to the current flow and other thermal factors. Key parameters include:

• \( \lambda \) - Thermal conductivity coefficient at zero electrical current.
• \( R \) - Electrical resistance.
• \( \zeta \) - Differential thermoelectric power.
  These generalized expressions help us understand more complex interactions between thermal and electrical processes, essential for technologies like thermocouples and thermoelectric generators.

thermoelectric effects

Thermoelectric effects refer to the direct conversion between thermal and electrical energy. They play a significant role in devices that generate electrical power from heat or provide cooling from electric power. Key thermoelectric effects include:

• Seebeck Effect: Generates an electric voltage across a material in response to a temperature difference.
• Peltier Effect: Causes heating or cooling at a junction of two different materials when current flows through the junction.
• Thomson Effect: Results in heating or cooling within a single conductor when electric current passes through it in the presence of a temperature gradient.
  These effects are harnessed in various applications, like thermocouples, which measure temperature, and thermoelectric generators, which convert waste heat into electricity. The generalized equations incorporating \( \lambda, \zeta, \) and \( R \) further expand on how these effects interplay within systems.

entropy balance equation

The entropy balance equation describes how entropy within a system changes over time due to various processes. It is expressed as: \[ \frac{\partial s}{\partial t} = -\abla \cdot J_{S} + \sigma \] where \[ J_{S} \] is the entropy current, and \[ \sigma \] is the entropy production. In a thermocouple, substituting for \[ J_{S} \] and simplifying yields: \[ \frac{\partial s}{\partial t} = \frac{\lambda}{T} \abla_{r}^{2}(T) - \abla_{r} \cdot \left(\frac{\Gamma}{T} I \right) + \frac{R I^{2}}{T} \] Each term on the right side represents different contributions to entropy production:

• The first term corresponds to thermal conductivity.
• The second term indicates entropy change due to thermoelectric effects (Peltier and Thomson).
• The third term represents Joule heating (resistive heating).
  This equation encapsulates how heat, electrical currents, and material properties contribute to changes in entropy within a system, providing detailed insights into the thermodynamics of non-equilibrium processes.