Contents

### Introduction

### 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*

### Specific Heat Capacity

**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**

*Specific heat capacity (SHC)***“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

**. 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:**

*ideal gas*This constant can be determined through experimentation, and is called the

**. 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:**

*Molar gas constant (R)***in a gas. However sometimes the information we are given, or we know, is the**

*number of moles (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:**

*number of mollecules (N)*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

**of the velocities. This is worked out in the following way:**

*root mean square*#### 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

**velocity explained above. Multiplying the square of the rms velocity by**

*root mean square***the number of molecules gives us our sum of the velocities in the equation above. This simplifies to the following:**

*N*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