Peltier Coefficient Unit Explained

The Peltier coefficient, often denoted by $\Pi$, is a fundamental property in thermoelectricity. It relates the heat absorbed or emitted at a junction between two different conductors to the electric current passing through that junction.

There are two common ways to define the Peltier coefficient, leading to its unit:

Method 1: Heat flow rate per unit current

The heat flow rate $\frac{dQ}{dt}$ at a junction is proportional to the current $I$ flowing through it. The constant of proportionality is the Peltier coefficient $\Pi$:

$\frac{dQ}{dt} = \Pi I$

From this relationship, the Peltier coefficient can be expressed as:

$\Pi = \frac{dQ/dt}{I}$

Now let's determine the unit using this formula:

• The unit of heat flow rate ($\frac{dQ}{dt}$) is power, which is Joules per second (J/s).
• The unit of current ($I$) is Ampere (A).

So, the unit of $\Pi$ is $\frac{\text{J/s}}{\text{A}} = \frac{\text{J}}{\text{s} \cdot \text{A}}$.

While correct, this unit is not one of the options provided. Let's consider the alternative definition.

Method 2: Heat per unit charge

Alternatively, the Peltier coefficient can be defined as the amount of heat $Q$ absorbed or emitted when a unit amount of charge $q$ passes through the junction:

$Q = \Pi q$

From this relationship, the Peltier coefficient can be expressed as:

$\Pi = \frac{Q}{q}$

Now let's determine the unit using this formula:

• The unit of heat ($Q$) is Joules (J).
• The unit of charge ($q$) is Coulombs (C).

So, the unit of $\Pi$ is $\frac{\text{J}}{\text{C}}$, which is also written as $\text{JC}^{-1}$.

This unit, JC⁻¹, represents Joules per Coulomb. Let's compare this with the given options:

• Option 1: $\text{JC}^{-1}$ - Matches our derivation.
• Option 2: $\text{JA}^{-1}$ - Joules per Ampere. This is not the unit of the Peltier coefficient.
• Option 3: $\text{JV}^{-1}$ - Joules per Volt. This is equivalent to Coulombs (C), based on $Q=VC$ or $E=qV$ where $C=Q/V$ or $q=E/V$. This is not the unit of the Peltier coefficient.

Therefore, the correct unit for the Peltier coefficient is JC⁻¹.