Question

Explain why the units for entropy have a dependence on temperature- for example, why are the units \(\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}\) ?

Short answer

The units for entropy, \findOneJ} mol^{-1} \findOne,-1}, reflect its calculation from heat transfer per temperature, with heat transfer measured in Joules (\findOneJ}) and temperature in Kelvin (K), hence showing the direct dependence on temperature.

Step-by-step solution

Understanding entropy

Entropy is a measure of the number of specific ways in which a thermodynamic system can be arranged, commonly understood as a measure of disorder or randomness. The formula for entropy change (\findOne of formulas for calculating entropy change includes heat transfer at a constant temperature: \( \findOne{T} \findOne{S} = q_{\findOne{rev}} \) where \( S \) is the entropy, \( q_{\findOne{rev}} \) is the reversible heat transfer, and \( T \) is the temperature.

Analyzing the entropy formula

In the equation \( \findOne{T} \findOne{S} = q_{\findOne{rev}} \), if we solve for entropy change \( \findOne{S} \), we get \( \findOne{S} = \frac{q_{\findOne{rev}}}{T} \). Here, \( q_{\findOne{rev}} \) has units of energy (Joules in the SI system), and \( T \) has units of temperature (Kelvin in the SI system). Therefore the units for entropy, \( \findOne{S} \), come from the division of these two quantities.

Deriving the units of entropy

Since entropy change is calculated as energy divided by temperature, the units naturally become \( \frac{J}{K} \), where Joules measure energy and Kelvin measures temperature. When considering the amount of substance as well, we include moles, leading to the units of entropy as \( \frac{J}{mol \findOne{K}} \). This shows the direct dependence of entropy on temperature.

Key concepts

Understanding a Thermodynamic System

In the context of thermodynamics, a system is a specific portion of space or a collection of matter which is being studied. All thermodynamic processes occur within this system, and it is categorized based on its interactions with the surroundings. There are three main types of thermodynamic systems:

• Isolated system: Exchanges neither matter nor energy with the surroundings.
• Closed system: Exchanges energy but not matter with its surroundings.
• Open system: Can exchange both energy and matter with the surroundings.

Understanding the concept of a thermodynamic system is crucial because entropy, a core concept in thermodynamics, is a property describing the state of this system. The entropy of a system gives us insight into the directional flow of energy and the extent of energy dispersal within the system.

The Process of Reversible Heat Transfer

Reversible heat transfer is a cornerstone concept in understanding how a system exchanges heat. It is an idealized process that assumes heat can be transferred between a system and its surroundings without increasing the total entropy of both.

In a real-world scenario, all processes cause the entropy to increase, making them irreversible. However, the concept of reversible processes is important because it establishes a standard by which real processes can be evaluated. To achieve reversible transfer, the system must change in infinitesimally small steps, maintaining equilibrium and allowing the heat transfer to occur at a constant temperature. This ideal rearview is crucial when we apply the entropy change formula since we assume reversible conditions to simplify the calculations.

Breaking Down the Entropy Change Formula

The entropy change formula, expressed as \( \Delta S = \frac{q_{\text{rev}}}{T} \), is a fundamental equation in thermodynamics. Entropy (\( S \)) represents the level of disorder within a system, and the formula allows us to calculate the change in entropy (\( \Delta S \)) resulting from a reversible heat transfer (\( q_{\text{rev}} \)).

Here’s why the formula is set up this way:

• The numerator, \( q_{\text{rev}} \), is the quantity of heat transferred reversibly and has the units of energy, typically joules (\( J \)).
• The denominator, \( T \), represents the absolute temperature in kelvin (\( K \)) at which the transfer occurs.
• When dividing these, the resulting units for entropy change are joules per kelvin (\( \frac{J}{K} \)).

When the amount of substance is also taken into consideration, the unit becomes joules per mole per kelvin (\( \frac{J}{mol\cdot K} \)), which provides us with a more comprehensive quantity for specifying entropy changes for a particular amount of a substance.