Equation of State in Thermodynamics

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Introduction

“The Adventure of the Dancing Men” is a short story written by Sir Arthur Conan Doyle, and it was originally presented in the collection “The Return of Sherlock Holmes”. In it, our hero is presented with a mystery involving drawings of dancing men.

The client is frantic with worry - he has been married only a year to a woman with a mysterious past, and when the drawings begin appearing in variegated places such as a door, wall, or windowsill, she is terrified. Holmes of course, being a cryptographic expert, quickly realizes that the dancing men are being used to transmit intimidating messages to his client’s wife. However, by the time he cracks the code, the denouement has come too quickly, with unexpected consequences.

Sometimes the language and symbols used in physics and engineering reminds one of cryptograms. One may feel like a Sherlock Holmes in trying to decipher all of the various meanings. However, people who practice cryptography deliberately attempt to obfuscate meanings, whereas the goal in science and engineering is to provide shortcuts that make communication easier. One such term which causes confusion and strikes fear into the hearts of beginners is “equation of state” and the other side of its Janus-like head, “state variables”. However, these phrases are not as complicated as they sound.

The Meaning of the Term “Equation of State” in Thermodynamics

The expansion of a gas is much more complicated than the expansion of a solid or a liquid. This is because gases are able to expand to fill whatever container you put them in.

The volume of a gas depends on pressure. However, it also depends on the temperature. As if that is not enough, we also need to consider its mass. Therefore, in order to develop a proper mathematical model of a gas, we need to consider its volume, pressure, temperature, and mass. We need an equation of state.

A relation between the physical terms pressure P, volume V, and temperature T is called an equation of state. State simply means the physical condition of the system. Here, we are using parameters to specify the physical relations we are interested in. The parameters P, V, and T are called state variables.

The physical state of our gas system can change. On a very hot day you may be reluctant to turn on the oven inside your kitchen, especially if you have no air conditioning. The heat from the oven makes the temperature of the air inside the kitchen increase, and this may make it unbearably hot inside your entire efficiency apartment. Turn the oven off, and the temperature will stop increasing. When the state or physical condition of our system changes, and values such as the pressure and temperature to stop increasing or decreasing, we have reached an equilibrium state. Our observed parameters will be the same throughout the system, and on the whole will not change in time.

States in Other Areas

The terms “state” and “state variables” are not confined to thermodynamics; in cosmology, the equation of state involves the ratio of pressure to energy density in dark or vacuum energy, and it is simply the cosmological constant w. In quantum mechanics, the quantum state of a microscopic particle may be modeled using functions with the variables position (space) and time.

So one does not need to morph into Sherlock Holmes, or even to hire one in order to decipher unfamiliar terminology in science and engineering. Patience is indeed the key.

Resources

Bright Hub Article on the Definition of Thermodynamics

References

Physics for Scientists & Engineers, Douglas Giancoli