Thermal Physics

Internal Energy and Temperature

The internal energy of an object this the total energy of it’s mollecules due to their motion and positions (potential energy). In some cases the internal energy of a material can be affected by other factors. For example magnetised iron has more internal energy then non-magnetised iron.

The internal energy of an object can be increased by:

  • Energy transfered to the object (via heat etc)
  • Work done on the object to increase potential energy

This brings us on nicely to the first law of thermodynamics:

The change of internal energy of a system is equal to the heat added to the system, minus the work done by the system

This is basically the principle of conservation of energy applied to thermal processes.

Specific Heat Capacity

The Specific heat capacity (SHC) of a material is the energy needed to raise the temperature of a mass of 1kg by one Kelvin. This is a charactieritic of materials, and varies from material to material. The letter used to denote the S.H.C is “c”. The following equation expressed the energy change required for a temperature change for a given SHC and mass:

Changes of State + Latent Heat

Experimental Gas Laws

The Ideal Gas Law

To get the ideal gas laws we combine all three of the experimental gas laws. Any gas which obeys Boyle’s law is considered an ideal gas. In reality no gas in the universe is a perfect ideal gas, however at low temperatures and atmospheric pressure, it is a good model for how many gases behave. Combining all three experimental laws gives the following:

This constant can be determined through experimentation, and is called the Molar gas constant (R). However the ratio pV/T also depends on the number of moles (n) in the gas. Therefore the ideal gas equation can be expressed in the following way:
This law can be expressed in a different way. The equation above is in terms of the number of moles (n) in a gas. However sometimes the information we are given, or we know, is the number of mollecules (N) of gas. We can therefore re-write the ideal gas law in terms of this instead. We know that the number of moles in a substance is the number of molecules divided by Avogadro’s number, therefore we can substitute this into our ideal gas law:

This gives us the following:

The Kinetic theory of Gases

Molecular Speeds

The velocities of molecules in an ideal gas are not all the same. In fact they are random, and distributed via the Maxwell-Boltzmann distribution. As the temperature of a gas increases the shape of the distribution changes because more heat means more energy for the molecules. The graph below shows how the velocity distribution changes with temperature:

When considering the velocity of the molecules in calculations an average is used. The average used in the root mean square of the velocities. This is worked out in the following way:

Deriving the Equations of Kinetic Theory

We want an equation which describes the movement of molecules in an ideal gas generally. Consider one molecule contained in a cubic container with dimensions (l,l,l). This molecule will have a component of it’s velocity in each of the 3 dimensions (x,y,z):

Let’s consider only the x direction, to simplify our calculations. Before a collision with the back wall the molecule’s momentum is equal to the following:

Assuming every collision with a wall is elastic, after the collision the direction of the velocity would have reversed, but the magnitude will stay the same, therefore the velocity after a collision with a wall is:

This means that the change in momentum after a collision is:

The time between successive collisions with a wall is twice the width of the box over the velocity of the molecule:

The force on the molecule is the change in momentum per unit time. Thus the force on the molecule after a collision is:

This allows us to work out the pressure on the container from the molecule:

This is the pressure on the container from just one of the molecules in the x direction, to get the total pressure in the x direction, we need to sum the pressures of all the molecules. Therefore for N molecules the total pressure in the x direction is:

We can factorise out the mass-volume ratio to simplify the sum:

We arbitrarily chose to start this derivation in the x direction, but the same thing can be done to the velocity of the molecule in the other two dimensions:

To work out the total pressure on the container from the gas, we need to average the pressures in each of the three dimensions:

Collecting the separate x,y and z components of each molecule’s velocity will allow us to simplify the equation:

The magnitude of a molecules velocity is given by the following (the resultant vector):

So we can substitute in each molecules velocity, for its velocity separated into the 3 dimension:

This is similar to the root mean square velocity explained above. Multiplying the square of the rms velocity by N the number of molecules gives us our sum of the velocities in the equation above. This simplifies to the following:

Giving us the final kinetic theory equation:

Assumptions for Kinetic Theory

For the derivation above to be viable, the following assumptions must be made:

  • The molecules of gas are point molecules, therefore their volume is negligible compared to the volume of the gas
  • The molecules aren’t attracted to each other
  • The molecules move in continual random motion
  • The collisions between the molecules and between the molecules and the container are elastic collisions
  • Each collision with the container is much shorter in duration then the time between successive collisions

Kinetic Energy