User:Seankieran/sandbox

Thermodynamics
In the most general case the Diet Coke and Mentos reaction can be represented by the equation:
 * $$mh_i + VP_i\ = mh_f + VP_f + \frac{1}{2}mv^2 + mgz + Q_{loss}$$

Where:
 * $$V$$ is the volume of the bottle
 * $$P_i$$ is the pressure due to the reaction of aqueous carbonic acid into gaseous carbon dioxide
 * $$P_f$$ is atmospheric pressure
 * $$v$$ is the velocity of the Diet Coke out of the bottle
 * $$z$$ is the maximum height achieved by the Diet Coke eruption

Considering that the overwhelming majority of the reaction is caused from the nucleation releasing pressure from inside the Diet Coke, we can safely neglect the contribution from the enthalpy change during the reaction. The minimal release of energy due to the reaction means that the only significant contribution to heat loss will be friction, which will be relatively small. This means that heat loss can also be ignored. The equation can then be simplified into:
 * $$ V \Delta P = \frac{1}{2}mv^2 + mgz$$

Defining the second state as the moment the Diet Coke begins erupting, the height will not have changed, allowing the maximum velocity to be defined as:
 * $$ v_{max} = \sqrt{\frac{2V\Delta P}{m}} $$

Alternatively the maximum height can be represented by the equation:
 * $$ z_{max} = \frac{V\Delta P}{mg} $$

since the velocity is zero at the peak of the eruption.