# Thermal Physics and Differentials

• bobpeg123
In summary, the conversation discusses the equation of state for a hypothetical substance with compressibility k = a/V and volume expansivity B = 2bT/V, where a and b are constants and V is the molar volume. The question asks to show that the equation of state is V = bT^2 - aP + constant, but the individual is uncertain about how to approach the problem and the definitions of compressibility and expansivity. They are advised to refer to their textbook or notes for a better understanding of the problem.

#### bobpeg123

A hypothetical substance has a compressibility k = a /V and a volume expansivity
B = 2bT /V , where a and b are constants and V is the molar volume. Show that the
equation of state is:
V = bT2 - aP + constant

To be honest I'm not entirely sure what I'm actually supposed to be doing with this question. Do I treat it as an ideal gas and therefore use that equation or is that completely wrong.

So far I have integrated each of the equations in respect to T, giving me,

(integral of) a/V dT = aT/V + c

(integral of) 2bT/V dT = bT^2/V + c

But now I'm stuck and can't seem to find a relevant relationship between P and what I've got there.

Your first equation is wrong. Compressibility has to do with the change in volume due to a change in pressure, not temperature.

So do I integrate both in terms of dP?

Both what?

What are the (mathematical) definitions of compressibility and expansivity?

Well I think they're k = 1/P and B = 1/T but I've been given equations for both in the question.

Sorry, I meant the general definitions in terms of derivatives.

To be honest I really don't know. I've become very confused with all the derivatives and general definitions.

Well, you should start by looking those up in your textbook or notes. If you don't know the basic definitions of the quantities involved in the problem, it's no surprise that the problem is confusing.

http://en.wikipedia.org/wiki/Material_properties_(thermodynamics [Broken])

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## 1. What is thermal physics?

Thermal physics is the branch of physics that deals with the study of heat and its effects on matter. It involves understanding the behavior of atoms and molecules in relation to temperature and energy.

## 2. How does heat transfer occur?

Heat transfer occurs through three main mechanisms: conduction, convection, and radiation. Conduction is the transfer of heat through direct contact between objects, convection is the transfer of heat through the movement of fluids, and radiation is the transfer of heat through electromagnetic waves.

## 3. What are the laws of thermodynamics?

The laws of thermodynamics are fundamental principles that govern the behavior of thermodynamic systems. The first law states that energy cannot be created or destroyed, only transferred or converted. The second law states that the total entropy of a closed system will always increase over time. The third law states that it is impossible to reach absolute zero (0 Kelvin) through any finite number of processes.

## 4. What is a differential in thermal physics?

In thermal physics, a differential is a small change or increment in a variable, such as temperature or energy. It is used in mathematical equations to represent the infinitesimal change in a quantity.

## 5. How is thermal equilibrium achieved?

Thermal equilibrium is achieved when two systems that are in contact with each other reach the same temperature. This occurs through the transfer of heat until the two systems have the same average kinetic energy of their particles. In other words, heat will flow from the warmer object to the cooler object until they are at the same temperature.