2. The First Law of Thermodynamics

2.1 The First Law

The first law of thermodynamics is the conservation of energy, including the equivalence of work and energy, and about assigning an internal energy to the system.

The first law of thermodynamics: When an amount of heat is supplied to the system and an amount of work is done by the system, changes are produced in the thermodynamic coordinates of the system such that

(2.1)

where is a function of the thermodynamic coordinates of the system. (In other words, is an exact differential.)

If the system is in adiabatic isolation, and . Since is an exact differential, this means that

(2.2)

Thus the adiabatic work done by the system is independent of the process involved and depends only on the initial and final states of the system. It is the recognition of this fact through careful experiments (by Joule) which led to the first law.

The quantity is called the internal energy of the system. For a gas, where the work done is given by (1.1), we may write the first law as

(2.3)

If we put , we get a relation between p and which is valid for adiabatic processes. The curve connecting p and so obtained is called an adiabatic. Starting with different initial states, we can get a family of adiabatics. In general, when we have more thermodynamic coordinates, adiabatics can be similarly defined, but are higher dimensional surfaces.

Specific heats

When heat is supplied to a body, the temperature increases. The amount of heat needed to raise the temperature by is called the specific heat. This depends on the process, on what parameters are kept constant during the supply of heat. Two useful specific heats for a gas are defined for constant volume and constant pressure. If heat is supplied keeping the volume constant, then the internal energy will increase. From the first law, we find, since ,

(2.4)

Thus the specific heat may be defined as the rate of increase of internal energy with respect to temperature. For supply of heat at constant pressure, we have

(2.5)

Thus the specific heat at constant pressure may be taken as the rate at which the quantity increases with temperature. The latter quantity is called the enthalpy.

In general the two specific heats are functions of temperature. The specific heat at constant volume, , can be calculated using statistical mechanics or it can be measured in experiments. can then be evaluated using the equation of state for the material. The ratio is often denoted by .

2.2 Adiabatic and isothermal processes

Among the various types of thermodynamic processes possible, there are two very important ones. These are the adiabatic and isothermal processes. An adiabatic process is one in which there is no supply of heat to the body undergoing change of thermodynamic state. In other words, the body is in adiabatic isolation. An isothermal process is a thermodynamic change where the temperature of the body does not change.

The thermodynamic variables involved in the change can be quite general; for example, we could consider magnetization and the magnetic field, surface tension and area, or pressure and volume. For a gas undergoing thermodynamic change, the relevant variables are pressure and volume. In this case, for an adiabatic process, since ,

(2.6)

From these, we find

(2.7)

Generally can depend on temperature (and hence on pressure), but if we consider a material (such as the ideal gas) for which is a constant, the above equation gives

(2.8)

This is the equation for an adiabatic process for an ideal gas.

If we consider an isothermal process for an ideal gas, the equation of state gives

(2.9)

2.3 Barometric formula, speed of sound

Here we consider two simple examples of using the ideal gas law and the formula for adiabatic expansion.

First consider the barometric formula which gives the density (or pressure) of air at a height h above the surface of Earth. We assume complete equilibrium, mechanical and thermal. The argument is illustrated in Fig. 2.1. We consider a layer of air, with horizontal cross sectional area A and height . If we take the molecules to have a mass m and the number density of particles to be , then the weight of this layer of air is . This is the force acting downward. It is compensated by the difference of pressure between the upper boundary and the lower boundary for this layer. The latter is thus , again acting downward, as we have drawn it. The total force being zero for equilibrium, we get

(2.10)

Thus the variation of pressure with height is given by

(2.11)

Let us assume the ideal gas law for air; this is not perfect, but is a reasonably good approximation. Then and the equation above becomes

(2.12)

The solution is

(2.13)

This argument is admittedly crude. In reality, the temperature also varies with height. Further, there are so many non-equilibrium processes (such as wind, heating due to absorption of solar radiation, variation of temperature between day and night, etc.) in the atmosphere that (2.13) can only be valid for a short range of height and over a small area of local equilibrium.

Our next example is about the speed of sound. For this, we can treat the medium, say, air, as a fluid, characterized by a number density ρ and a flow velocity . The equations for the fluid are

(2.14)

The first equation is the equation of continuity which expresses the conservation of particle number, or mass, if we multiply the equation by the mass of a molecule. The second term is the fluid equivalent of Newton’s second law. In the absence of external forces, we still can have force terms; in fact the gradient of the pressure, as seen from the equation, acts as a force term. This may also be re-expressed in terms of the density, since pressure and density are related by the equation of state.

Now consider the medium in equilibrium with no sound waves in it. The equilibrium pressure should be uniform; we denote this by , with the corresponding uniform density as . Further, we have in equilibrium. Now we can consider sound waves as perturbations on this background, writing

(2.15)

Treating and as being of the first order in the perturbation, the fluid equations can be approximated as

(2.16)

Taking the derivative of the first equation and using the second, we find

(2.17)

Where

(2.18)

Equation (2.17) is the wave equation with the speed of propagation given by . This equation shows that density perturbations can travel as waves; these are the sound waves. It is easy to check that the wave equation has wave-like solutions of the form

(2.19)

with . This again identifies cs as the speed of propagation of the wave.

We can calculate the speed of sound waves more explicitly using the equation of state. If we take air to obey the ideal gas law, we have . If the process of compression and decompression which constitutes the sound wave is isothermal, then .

Actually, the time-scale for the compression and decompression in a sound wave is usually very short compared to the time needed for proper thermalization, so that it is more accurate to treat it as an adiabatic process. In this case, we have , so that

(2.20)

Since , this gives a speed higher than what would be obtained in an isothermal process. Experimentally, one can show that this formula for the speed of sound is what is obtained, showing that sound waves result from an adiabatic process.