Microstate and Macrostate in Thermodynamics
In this class, you will learn to differentiate microstates and macrostates in thermodynamics, exploring how the microscopic configurations of a system determine its macroscopic properties. Through concrete examples, such as the calculation of combinations in systems with multiple possible states, you will understand key concepts that connect the microscopic with the observable. Discover how these ideas are fundamental to interpreting the behavior of complex thermal systems and develop a new perspective on the principles governing matter.
Learning Objectives:
By the end of this class, the student will be able to:
- Distinguish between microstate and macrostate, understanding their definitions and applications in thermodynamics.
- Classify microstates into macrostates based on common qualities and calculate their number using combinatorics.
- Explain how the equiprobability of microstates determines the probability of macrostates in thermodynamic systems.
- Relate the microscopic properties of a system to its macroscopic properties such as pressure, volume, and energy.
- Analyze thermal systems through practical examples, highlighting the relevance of microstates and macrostates.
TABLE OF CONTENTS:
The distinction between microstate and macrostate in thermodynamics
Qualities of macrostates in thermodynamics
The distinction between microstate and macrostate in thermodynamics
An essential distinction in thermodynamics involves the difference between microstate and macrostate. It is generally said that the microstate is related to the microscopic aspects of matter, while the macrostate deals with the entirety of a system composed of many micro-systems. While this distinction helps guide initial understanding, it does not fully capture the essence of microstate and macrostate.
To gain a more precise understanding of these concepts, let’s examine the following example:
Microstates and macrostates with coins
Imagine you have a box with 100 coins that remains closed at all times. If you shake this box vigorously, the coins will tumble inside and land in some position: heads or tails. Consequently, the entire system has a vast number of possible states it could settle into: 2^{100} in total (try estimating the order of magnitude of that number!). Each configuration of heads and tails is referred to as a state. Assuming all these configurations are equally probable, each has a probability of 1/2^{100} of occurring.
Each specific configuration is a microstate
We define each specific configuration as a microstate of this system. For example, the first coin lands heads, the second tails, the third heads, and so on. Identifying a microstate requires pinpointing each individual coin, which in practice can be cumbersome.
A macrostate is the family of all microstates sharing a common quality
At this point, instead of studying each microstate individually, we can classify the possible microstates into families with a common quality: those with one head, those with two heads, those with three heads, and so on. Using combinatorics, we find that the microstates with:
- 0 heads and 100 tails are: \displaystyle{{100}\choose{0}} = \frac{100!}{(100 - 0)! 0!} = 1 state in total
- 1 head and 99 tails are: \displaystyle{{100}\choose{1}} = \frac{100!}{(100 - 1)! 1!}= \frac{100!}{99!} = 100 states in total
- 2 heads and 98 tails are: \displaystyle{{100}\choose{2}} = \frac{100!}{(100 - 2)! 2!}= \frac{100!}{98!2!} = \frac{99\cdot 100}{2} = 99 \cdot 50 = 4950 states in total
- 49 heads and 51 tails are: \displaystyle{{100}\choose{49}} = \frac{100!}{(100 - 49)! 49!}= \frac{100!}{51!49!}\approx 9.89\cdot 10^{28} states in total
- 50 heads and 50 tails are: \displaystyle{{100}\choose{50}} = \frac{100!}{(100 - 50)! 50!}= \frac{100!}{50!50!}\approx 10^{29} states in total
Thus, the macrostate “0 heads” contains a single microstate, the macrostate “1 head” contains 100 microstates, the macrostate “2 heads” contains 4950 microstates, and so on.
Qualities of macrostates in thermodynamics
In general, a macrostate consists of a large number of equally probable microstates.
Although all microstates are equally probable, macrostates are not. The probability of a macrostate is proportional to the number of microstates it contains. The most probable macrostate is the one with the greatest number of microstates.
Thermal systems behave in a very similar way to the example just reviewed. Specifying a microstate in a thermodynamic system requires indicating the magnitudes that define a certain configuration of the system at the microscopic level, such as the position, velocity, or energy of the atoms comprising the system.
In practice, it is impossible to measure each of the system’s microstates. On the other hand, macrostates can be described solely in terms of the macroscopic properties of the system, such as pressure, total energy, or volume. A macroscopic configuration of a system with 2[m^3] of volume at a pressure of 35[kPa] may be associated with a large number of microscopic configurations.