# First Law of Thermodynamics conversion

I went to a random page in my physics textbook and came across the First Law of Thermodynamics topic. I am intrigued by the simplicity of the mathematical statement of this law, which is dU = dQ - dW. (dU is the small change in internal energy, dQ is the small change in heat, and dW is the small amount of work done by the system). The very first verbal description of the First Law of Thermodynamics I read in the physics textbook is that this Law is a principle of conservation of energy, that energy cannot be destroyed and created but converted from one form to another. My question is how does that description (definition?) translate to the mathematical statement dU = dQ -dW.

It's actually quite simple. In thermodynamics there are two types of energy: That which can be used to do useful work (like the thermal energy of gas particles pushing a piston), and that which cannot be used to do useful work. The latter is called heat. So when there is a change in the energy of the system, it lost/gained heat and/or did work. This is fairly simplistic but I think you'll get the point.

turin
Homework Helper
I must disagree with nealh149. The only energy that appears in the law is the so-called internal energy of the system, U. This energy can be "used" for both work and heat, and this is not an issue of the "type of energy". In fact, the first law of thermo very profoundly suggests that exactly the same type of energy (U) can be transfered either in work or heat. Heat is not a type of energy, it is a mode of energy transfer, as distinguished from work.

BTW, you should be careful of the minus signs that appear in the mathematical version of the first law of thermo. The signs have meaning, but they depend on the definition. For example, if U is the internal energy of the system, Q is the heat flow into the system, and W is the work done by the system, then dU = dQ - dW. This simply means that energy conservation is accounted by increasing the internal energy in the exact amount that was transfered into the system by heat, less the amount of energy that was required by the system to do work.

In summary, the first law of thermo says that there is one kind of energy, and two ways that energy can be exchanged between systems.

It is actually the second law of thermo that is responsible for restricting the availability of the energy to do mechanical work.

OK, thank you for you replies... but I'm not sure if I truly understand your posts.

I think the confusion I have with this whole issue is how does the mathematical statement dU = dQ - dW imply that energy cannot be created or destroyed, only changed from one form to another.

I reread the section in my my textbook and it states that the first law of thermodynamics is a conservation of energy statement for thermodynamic systems that exchange energy with their surroundings.

turin says that the First Law states that there is one kind of energy and there are two ways that energy can be exchanged between systems.

This is my personal understanding of the problem. Let's think about this issue in a conceptual way and with an example. U is the internal energy, Q is the heat that is transferred in and out of the system, W is the work expended by the system. So if we have a piston (as used in the textbook examples), Q enters the piston causing the molecules in the piston to expand. Work can be obtained by how much the piston expands (since work is equal to the amount of energy causing movement in the air molecules). So Q comes into the system (piston) and it causes the air molecules to move (in the form of work, kinetic energy, etc...). If Q>W, then not all of the heat is used up and there will be a positive U. U is the energy that hasn't been used up to heat the molecules that is leftover.

To summarize the paragraph above, U is the amount of energy leftover in the system after Q enters the system and is used up in the form of W.

I guess that since the mathematical statement dU = dQ - dW DOES imply that energy is neither lost or created and only converted into one form or another, since all energy is accounted for in the equation and there is no net gain/loss of energy in the equation.

While I was trying to covey what turin said, he said it much better.

Andy Resnick
I went to a random page in my physics textbook and came across the First Law of Thermodynamics topic. I am intrigued by the simplicity of the mathematical statement of this law, which is dU = dQ - dW. (dU is the small change in internal energy, dQ is the small change in heat, and dW is the small amount of work done by the system). The very first verbal description of the First Law of Thermodynamics I read in the physics textbook is that this Law is a principle of conservation of energy, that energy cannot be destroyed and created but converted from one form to another. My question is how does that description (definition?) translate to the mathematical statement dU = dQ -dW.

There's a couple of points worth mentioning. First, the expression dU = dQ - dW is written for a system that is somehow differentiated (perhaps by a material boundary) from the external environment. dW and dQ then represent flows of energy into and out of the system, and there are two idealized forms of energy: so-called 'workless dissipation' and 'dissipationless work'. Heat flow (dQ) is thought of as workless dissipation- energy that does no work (i.e. heat)- and is related to the entropy. 'Work', dW, is usually introduced as P dV type mechanical work, but 'work' can take many forms-creation of new interface (surface tension), chemical reactions (chemical potential), electromagnetic work, etc. etc.

So, the statement dU = dQ - dW is simply a statement saying that the change of (internal) energy in a system is equal to the amount of energy coming into the system or flowing out from the system into the external environment. This is another way of saying energy is conserved.

Nice ....

If I were the scientist who authored the First Law of Thermodynamics, why would I have the change in internal energy equal heat (in/out of the system) minus work as the law? How is the law derived? What is the significance of that statement dU = dQ - dW? I apologize if you might have answered these questions already in previous posts, but I want to make sure that I 100% get the answer since I still doubt whether I truly understand it.

turin
Homework Helper
See the wikipedia article on the "Mechanical Equivalent to Heat".

The first law of thermodynamics predates our atomic theory of matter. So, at the time, it was a big discovery and it was a law, which in old physics, means it was not derived, simply stated from experimental intuition (like newton's LAW of universal gravitation). However, given our modern understanding that matter is in fact made of atoms the first law is actually a trivial statement of conservation of energy since we now understand that temperature is simply the average kinetic energy of the particles of a system. Therefore, the first law essentially says, in a modern context, that the net change in energy in a system is equal to the energy you added to the system minus the energy you took out of the system. Which, if energy is conserved, is kinda a 'no duh' statement (although for the time it was a great discovery)

Andy Resnick
Nice ....

If I were the scientist who authored the First Law of Thermodynamics, why would I have the change in internal energy equal heat (in/out of the system) minus work as the law? How is the law derived? What is the significance of that statement dU = dQ - dW? I apologize if you might have answered these questions already in previous posts, but I want to make sure that I 100% get the answer since I still doubt whether I truly understand it.

The law of conservation of energy is not derivable; it's a postulate. The expression dU = dQ - dW as a statement of the law was first written down when we only knew about two kinds of energy- by that I mean to say back then, if I built a machine, the machine did a combination of two things: move something and heat up. So, the two terms reflect what was known then.

Nowadays, we have a much better understanding of 'energy', the different forms it takes and how it flows. However, we do not need to change the expression dU = dQ - dW, we simply say that the flow of energy can produce 'work' (which can be mechanical, chemical, electromagnetic, interfacial...) or 'heat' (which we don't understand as well, but gets tied into the entropy and dissipation).

Does that help?

infomax
I went to a random page in my physics textbook and came across the First Law of Thermodynamics topic. I am intrigued by the simplicity of the mathematical statement of this law, which is dU = dQ - dW. (dU is the small change in internal energy, dQ is the small change in heat, and dW is the small amount of work done by the system). The very first verbal description of the First Law of Thermodynamics I read in the physics textbook is that this Law is a principle of conservation of energy, that energy cannot be destroyed and created but converted from one form to another. My question is how does that description (definition?) translate to the mathematical statement dU = dQ -dW.

Simply understand that the heat given to system will be equal to change in internal energy which changes temperature plus the energy utilized to work to displace the piston of the vessel(P.dV)
is to say that the Heat energy given to system will change the temperature of the system and do displacement of the piston. No energy is lost or destroyed,then first law hold good. 