Carnot and Otto Engines

It is possible to create a transfer of energy due to a difference in temperature. We saw this when a device used to transfer energy has one end place in cold water, and the other in warm water. The device has a disk that spins in whichever direction the heat flows.

Change in internal energy is equal to the addition of heat and work performed. Since heat equals the product between the number of moles, the molar heat capacity at constant pressure, and the change in temperature, and the work perform equals the product between the number of moles present, the ideal gas constant, and the change in temperature, these values can be substituted into the first law of thermodynamics, and a relationship between the ideal gas constant and the molar heat capacity at constant pressure is formed. 

Since the ideal gas constant equals the difference between the molar heat capacity at constant pressure and the molar heat capacity at constant volume, R can be substituted for. This forms a relationship between the product of the change in temperature and the number of moles present, and the  change of the product of pressure and volume over the difference in molar heat capacities and constant pressure and volume respectively.

For an adiabatic process, we determined that the sum of the change in pressure divided by the initial pressure, and the change in volume over the initial volume multiplied by gamma results to zero. Because gamma is always greater than one, the ratio of the change in pressure to initial pressure must be greater than the ratio of the change in volume to initial volume.

Through some algebra we arrive at this conclusions which states that for an adiabatic process, the product between the temperature and the volume raised to one less than gamma is constant.

This is the derivation for work in an adiabatic process.

When given the initial and final pressure and volume for an adiabatic process, finding the work done is easy.

From A to B, and C to D, the system undergoes an isothermal process From B to C, and from D to A, the system undergoes an adiabatic process. This is known as the carnot cycle.

This table list the changes in internal energy, as well as the work perform and heat associated with each process. We found that the efficiency of the above carnot cycle is 36%. This is the highest possible efficiency for an engine operating between 200 K and 300 K.

This machine demonstrates the process that an otto engine undergoes. It operates on a four stroke engine cycle. There is in intake stroke, a compression stroke, a power stroke, and an exhaust stroke. Air and fuel enters the cylinder on the intake stroke, and compressed in the compression stroke. The air fuel mixture is then ignited, violently forcing the piston down, then then piston pushes all the excess heat out of the cylinder, and the cycle repeats.

In a diatomic molecule, there are an extra 2 degrees of freedom. One is associated to the rotation perpendicular to the molecules axis, and the other perpendicular to each other.

Increasing the pressure as the piston compresses the gas, providing larger cylinders, and increasing the rate at which the cams operating the piston rotates can all increase the power output of an engine.