Work, Heat, Energy. and the First Law

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Note: This section needs a very basic knowledge of calculus. To see a simplified version go HERE


  • Understand the Concepts of Work, Heat, and Energy
  • Make some Observations of the nature of energy
  • Conclude from these the First Law of Thermodynamics



Force acting through a distance

Therefore, Work, W is

dW = − Fdl

where F is force and dl is the distance through which the force acts The reason for the minus sign is explained below.

In most applications of thermodynamics we are mainly interested in mechanical work due to pressure of a fluid. Since pressure is force per unit area, force is simply pressure times area:

F = − PA

If we consider a volume, V, then the distance l is






If we then assume a constant area then we can take the A inside the differential and:


Integrating gives



Capacity to do work
Internal Energy, U 
The total energy of the system[1]


Energy transferred due to a temperature difference
No heat transfer between a system and its surroundings
Exothermic process 
A process which releases heat
Endothermic process 
A process which adsorbs heat

Heat is denoted by the symbol, Q

Heat and work are not properties

It is important to note that heat and work are not intrinsic properties of a system. They refer only to energy which is transferred. We cannot say, for example, that a brick has 15 J of heat. It may however, have 15 J of energy.

Sign convention and notation

Heat and work are considered positive if they are transferred from the surroundings to the system.[2] This is the reason for the negative sign in the work equations above.

Δ is used to indicate finite change (for example, ΔU)

d is used to indicate differential change (for example, dU)

However, we do not use ΔQ or ΔW for finite changes in heat or work, since Q and W only refer to change. We but simply use just Q or W. We still do use dQ and dW for differential change.


The laws of thermodynamics are based on observations of the natural world. The first law is based on two observations concerning energy:

  1. Energy can be transferred between a system and its surroundings by only two ways: work and heat
  2. The total energy of a system and its surroundings is always constant (The conservation of energy)

First Law

These two observations can be combined into the First Law of Thermodynamics:

The internal energy of a system is constant unless changed by doing work or by heating

Mathematical Statement

Mathematically, the change in internal energy is the sum of the work and heat entering or leaving the system:

ΔU = Q + W


dU = dQ + dW


  1. Note that some references say the internal energy is the energy due to the internal vibrations, etc. In other words that other than kinetic or potential energy. However, since we are interested only in energy differences the definition used here is equivalent and is easier to understand.
  2. It is important to note that previously engineering used a different convention: Heat was the same, but work was considered positive if it was transferred from the system to the surroundings.
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