## J1jT [ dv p J i ds

I — I , and other thermodynamic diagrams, such as the (p-h) for refrigerants,

p dp p V dv for an ideal gas, this reduces to is also explained to clarify the physics involved;

dp dh

To obtain pressure-temperature relation for an ideal gas under reversible dp adiabatic conditions, we start with dT

 p, s s, p _ T, s _ _ T, s _

. Substituting for

. Substituting for the numerator and the denominator from Jacobian properties, then we get c

I ztL I =_L_= _P I —_ I which is a general result. Using the ideal gas

equation, the partial derivative on the right can be evaluated. Using the difference between the two specific heats for an ideal gas, as was shown above, the result be-

comes k=

Therefore,

dp / p where k is the ratio of the specific heats, k which upon integration gives the well-

T=p dT / T j k — 1 . This result is only valid if we have an ideal gas which undergoes a reversible and adiabatic, constant entropy, process. Again, this fact is not trivial and must be kept in mind before this result can be used. Similar results can be obtained for dv_ dT

dp in general, and when applied to an ideal

s c c v c gas would give f dv/V | = -|—1—| or \t = v1 k ]. For f — | we get the re-5 5 idT / T) [k -1) L J 13v)s 5

sult, in general, | — | = k(— | . This equals for an ideal gas | d / p | = -k

The total intent in all this was to verify the physics of the situation using mathematics. Mathematics is just a tool to totally explain these physical phenomena that characterize nature. Nature is what physics is; mathematics is the explanation mechanism and nothing more.

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## Renewable Energy 101

Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable. The usage of renewable energy sources is very important when considering the sustainability of the existing energy usage of the world. While there is currently an abundance of non-renewable energy sources, such as nuclear fuels, these energy sources are depleting. In addition to being a non-renewable supply, the non-renewable energy sources release emissions into the air, which has an adverse effect on the environment.

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