The simplest definition of the first law of thermodynamics is, “We can neither create energy nor destroy it. The energy can however be changed from one form to another”. This is the simplest yet elusive concept of the first law of thermodynamics. Before coming to a detailed explanation of the first law, we must know about “work” and “energy” first.

**Work**

The most fundamental definition of the work is, “The product of the force (applied) and the distance covered by the object”. There are many different kinds of work, like Electrical Work, Work due to Gravity, etc. However, the one we are going to discuss here is the expansion work.

Suppose we have a container filled with gas which has some initial volume. We apply some external pressure to the piston of the container, by pushing it through a distance “l”. The volume of the gas decreases. From the definition of pressure we know.

Pressure = F/A

⇒ F = P_{ext} A … (1)

Since work is the product of force and displacement

Here, displacement is the distance through which we’ve pressed the piston of the container. Therefore,

W = – Fl

Using equation (1)

W = -Fl = -P_{ext} Al

W = – P_{ext} ΔV

P_{ext} means, the surrounding is experiencing pressure on the system, not the system itself. Here is an important convention. The convention says if the environment (surrounding) pushes the piston, or in other words, if the system undergoes compression, the work is positive. We use the negative signs for the positive work.

If the system undergoes compression i.e. V_{2} <V_{1}, which means ΔV<0, which implies that:

ΔW= – (P_{ext} (-ΔV)).

⇒ ΔW> 0

**Work and its Path Dependency**

The above case defines work when we apply constant pressure on the system. However, we can change the pressure through infinitesimally small steps. This will give us a path along which both pressure and its corresponding volume vary. In mathematics, we use the differential form of the equation to describe instantaneous changes. Because we can’t switch between two extreme values of the volume in one step. So, the differential form of the work is:

đW = – P_{ext} dV . . . (2)

The little bar on “đ” represents that work is not an exact differential. An exact differential is one which can be defined by mere its initial and final positions. But in the case of thermodynamic or expansion work we have to define, how the current state is achieved, i.e. the path followed. Because work (expansion) is a path function. However “dV” is an exact differential, since it doesn’t care whatever path we follow between two different states. An exact differential has no interest in knowing the history through which the system went but an inexact differential does.

Also read: Zeroth Law of Thermodynamics

By integrating equation (2), we have:

W = ʃ đW = – ʃ Pext dV

Let us now discuss the path dependence of the work. Consider an ideal gas that has an initial state (P_{1}, V_{1}). We change its state and the final state is (P_{2}, V_{2}). We can bring this change in the state of our system through various paths, but we would discuss only two of them to grasp the concept.

Note: V_{1} > V_{2} & P_{1 }< P_{2}

The path in going from V_{1} to V_{2} is isobaric (constant pressure), and from P_{1} to P_{2} is isochoric (constant volume).

Let us assume that both the processes, shown above, are reversible, i.e. at any point we can go back without losing energy. The reversible processes are very slow in nature. We can calculate work for both these processes in the following way:

**For Path (1**)

**For Path (2)**

Here, P2> P1 which implies that W2> W1. It is evident from the above calculations that work done is dependent on the path we take.

Now we can obtain a thermodynamic cycle from the two paths discussed above. This will help us understand the concept of a simple thermodynamic cycle. By grasping this concept it would be quite easy for us to easily understand other complex cycles of thermodynamics.

In the figure above, we just added the two paths obtained previously. The total work is,

W_{Total} = W_{1} – W_{2} = P_{1}(V_{1}-V_{2}) – P_{2}(V_{1}-V_{2})

W_{Total} = (P_{1}-P_{2})(V_{1}-V_{2})

Here, P_{1}-P_{2} <0 (Since the pressure at point 2 is greater than the pressure at point 1)

Therefore, W_{Total }<0

The work is negative which means the system did work on the environment. The area inside the rectangle represents the work done by the system. The above system is just like a heat engine. Here is an important thing to consider, since we started from the initial state and reached back to the same initial state, but the work done is not zero. This is the definition of an inexact differential. If we had an exact differential the work done around a closed path would have been zero.

**Energy**

Now that we’ve defined the primary concept of the first law of thermodynamics which is “work”, there is a property that tells us about the capacity of a system to do work. We call this property “energy”. For example, a body at a higher altitude has more capacity to do work than the one at a lower altitude, due to its higher potential energy. So energy is nothing but the capacity of a system to do work.

**Internal Energy**

Let us perform an experiment. Suppose we have an adiabatic system. An adiabatic system is one in which no heat enters or leaves the walls. The walls are perfectly insulated. We have a flask in which there is some liquid filled. J.P. Joule performed the same experiment when he gave the concept of the first law of thermodynamics. We first churn the contents of the flask with the help of a paddle which is attached to a falling weight through a pulley. We place a thermometer inside the flask to read its temperature.

The paddle wheel starts rotating and churns the liquid as we drop the weight. The temperature inside the flask rises. We can calculate the work done as we know the weight. The state of our system has changed and we know the amount of work done to bring that change. Now we wait for the system to return to its initial state.

As the system returns to its initial state, we bring it back to the state we previously obtained with the help of falling weight. But this time we use an electric heater to do that. As we calculate the work, we come to know that the same amount of work has been done to bring the system to the same state. So the same amount of work brings in the same amount of change in the state of the system. Therefore, we assume that there is a property associated with the change in the state of the system and we call that property “internal energy”.

You can find this confusing with an already conceived definition of the work. But work done here is independent of the path since it is an adiabatic process.

**Heat**

Now the internal energy of a system depends on the work done. But the question is does it happen all the time?

Let’s go back to the experiment once again. But this time we remove the insulation and the system is no more adiabatic. We start churning the contents of the flask to bring it to the previously obtained state. What do we notice? It is taking more time to reach the same state, i.e. more work needs to be done on the system. We conclude that the internal energy of a system depends on another property besides work. We interpret this as, since the system has no insulated walls, therefore, it transfers energy to its surroundings due to the temperature difference.

The energy transfer that results due to the difference in temperature is “**heat**“. We can calculate heat by simply subtracting the amount of work done when the system was not insulated (to bring about the same change in state) from the amount of work done when the system had adiabatic walls.

**First Law of Thermodynamics**

We can put all this discussion in a single context which explains the relationship between the internal energy, work and heat. This relationship is what we call the “First Law of Thermodynamics”.

Mathematically,

ΔU = Q – W

“ΔU” represents the change in the internal energy of the system. The system here is closed. A closed system is one which allows heat and work transfer from its boundary but not the matter.

“Q” represents the amount of heat given to the system and “W” represents the work done by the system on its surrounding. This empirical statement of the first law of thermodynamics tells us that when the system does work its internal energy decreases. To compensate for that internal energy, we must supply heat to our system. In other words, we cannot attain perpetual motion.

To keep our system doing work, we must keep on supplying it with fuel (energy). Suppose we say, the system works at the expense of its internal energy (which is not possible), it would get exhausted in the end. Therefore, no system can defy the first law of thermodynamics.

**Energy Conservation and The First Law of Thermodynamics**

The first law of thermodynamics reveals a number of important and interesting facts. It says that any heat supplied to a system can be used in two ways:

- To increase its internal energy
- Work done by the system

Suppose we have a system and we provide it with heat. If the volume of the system doesn’t change, i.e. it is an isochoric process. Then we know that work done would be zero. Since W= PdV. This implies that all the energy that we have supplied to the system is consumed in raising its internal energy. That is the principle of the conservation of energy. Energy can neither be created nor destroyed, it just changes its forms. The practical examples which we discussed are clearly confirming the empirical statement of the first law of thermodynamics.

There were certain limitations with the first law when it first came into the discussions. Like, it didn’t tell us about the direction of heat flow, or the efficiency of a heat engine. However, these limitations have been addressed by the second law of thermodynamics.

If you want another perspective on understanding the philosophy behind the First law of thermodynamics you can visit Glenn Research Centre.