Gas Expansion at Constant Pressure

The initial pressure, which is held constant, is

`P=E_0/(2r_Py_(P0))`

where `E_0` is initial energy, `r_P` is cylinder width/2, and `y_(P0)` is the original height of the piston above the bottom of the cylinder. Then the energy expended by impacts on the moving piston is

`E=2r_PP_0(y_P-y_(P0))`

Suppose we just want to double the initial internal energy while letting the gas expand to yP1. We must supply heat energy `Q`

`Q=E_0+E_0(y_(P1)-y_(P0))/y_(P0)=E_0y_(P1)/y_(P0)`

where the expansion ratio is `e_r=y_(P1)/y(P0)` . The final internal energy is 2E0. For this animation I will define the heat capacity at constant pressure as the ratio of the heat input, including both the heat to increase the kinetic energy as well as the heat needed to negate the lost kinetic energy due to expansion, divided by the increase in kinetic energy of the discs. Here the gas expands because it is heated. The gas is held at constant pressure by the mass Force, `Mg`, of the piston where `M` is the piston mass and `g` is the acceleration of gravity. The gas expansion factor is proportional to the internal translational kinetic energy of the particles which, in turn, is proportional to the absolute Kelvin temperature of the gas. Therefore, the height of the piston is a measure of the gas temperature. If we heat the gas to two known temperatures and call that temperature difference 100 degrees we will have defined an absolute scale of temperature. If we extrapolate this scale to the case where the piston height would be zero, then we will have an estimate of absolute zero with respect to the two known temperatures. If the two known temperatures are the triple point (freezing temperature) of water and its boiling point at standard pressure, then this will result in a calibration of the Celsius scale which has a zero of -273 degrees C. We want to determine how much heat energy, for each expansion increment, needs to be added to keep the gas at constant pressure while also increasing the internal kinetic energy of the gas. For a simple 2D gas like we have here, it is shown that the heat energy, `Q`, needed for each increment of internal energy, E, is

`C_pE=(dQ)/(dE)=2`.

This expression is completely analogous to the heat capacity at constant pressure which is usually stated as:

`C_pT=(dQ)/(dT)=2nR`

where `T` is absolute (Kelvin) temperature, `n` is the number of moles in the sample and `R` is the molar gas constant. But `C_pE` has the advantage that its dimensions are a simple ratio and does not introduce artificial and hard to define values like absolute temperature. The best way to demonstrate the gas internal energy change due to the piston rising or falling is to apply heat energy to the gas instantaneously and then observe the average gas energy as piston rises or falls. The step increase in internal energy due to the heat will result in a temporary pressure increase. The force from this pressure increase will cause the piston to move upward and come to a new equilibrium height (or volume). As the piston rises, the gas internal energy (and temperature) will drop by a certain fraction, `(C_p-C_v)/C_p`, of the internal energy increase where, `C_p`, the heat capacity at constant pressure is `5/2R` for monatomic gases and `C_v`, the heat capacity at constant volume is `3/2R` for monatomic gases. Then `C_p-C_v=R` for all gases where `R` is the molar gas constant. Thus the reduction fraction for monatomic gases is `2/5`. This reduction is shown both by a plot of `U` Vs time as well as a change of the color temperature of the gas. When heat is removed instantaneously from the gas, the piston moves downward to its new equilibrium height and the gas energy is increased by the change in gravitational energy of the piston's mass. This is also shown by a plot of the internal energy Vs time and an increase of the color temperature of the gas. The learner may (within limits) use the Sliders to change the following variables:
1. Initial average Energy of atoms
2. Radius of atoms
3. Total Number of Atoms
4. Piston Speed
5. Expansion Ratio