If your book still has Heat Capacity Ratio, the methodology printed is likely nonsense.

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There are several ways to measure or calculate Beta and Gamma. Heat Capacity Ratio is not nonsense, but most methodology descriptions are incorrect. It can only be calculated by the "specific heat" method, if you actually measure the amount of heat required to do both constant volume heating and constant pressure heating, to and from the same temperature It can also be calculated from two points on the Work Curve, using ratio of logarithms. It can be calculated between any two points on the isobaric curve (a flat line of pressure vs volume), by doing simple division of the relative measured work heat and specific heat. Work heat can be calculated by subtracting the specific heat from the quantity of heat required to do isobaric volume change, which is the sum of specific heat and work heat. It can not be calculated between any point on the work curve and a point on a different curve, such as isothermal or isobaric. Heat to constant volume requires the same work (0) for all temperature/pressure changes, so is a pure measurement of specific heat. Heat at constant pressure or constant temperature or any other curve, can take anywhere from 0 work to infinite amount of work. So the "ratio" could be anywhere between 1 and infinity, depending on how much the volume changes.

Measuring Specific Heat and Work Energy

Beta is the ratio of work energy to heat energy for a given volume and pressure. Or one could say Beta is the work/heat-capacity ratio, that is required for a vapor to occupy the same volume and pressure.

It is Not the ratio of anything and the specific heat under constant pressure, because specific heat does not change under constant pressure or constant volume. When volume is allowed to change in order for pressure to stay the same, work is done to expand the volume at that pressure, which takes the same amount of energy for all possible gases (force times distance or pressure times volume change). The heat capacity of a vapor (gas) is (about) the same at all volumes, temperatures and pressures (provided it remains in the gaseous state, the ideal gas law applies). Most current texts either leave this out, or explain how to arrive at it by taking some ratio of vapor heated under constant volume and then letting it expand vs letting it expand as it is heated. This method cannot arrive at Gamma or Beta, because the measurement of Work Energy (Heat) is skipped. The value of the ratio of anything from the constant-pressure volume to the constant-volume volume varies from 0 at very small volume change to infinity over vastly differing volumes or pressures.

Beta of a gas is the Work Energy divided by the total heat stored in the gas. For air, beta is about .4, meaning 1 unit area takes .4 heat units (implicitly using 1 heat unit is ambient temperature at ambient pressure). Taken the other way, 1 heat unit will expand over 2.5 (1/0.4) Work units (unit squares of Pressure x Volume).

And to get Gamma add 1 to Beta.

 

Calculating Beta from Pressure Measurements

It is indeed difficult to precisely measure the amount of heat added to a body of gas, to precisely confine gas or precisely allow expansion under a precise constant pressure. Everything about the first way Gamma was measured is difficult to do, much less to 3 decimal places.

We have the luxury of knowing the Pressure curve is an exponential function of Volume. So a much simpler way to arrive at Gamma is to calculate it from pressure measurements and volume measurements. These measurements are static, and do not depend on timing or rate of heat added or even having separate vessels for measurements.

Start with a volume of the vapor in a piston/cylinder pair. Measure the initial volume and pressure.

Change the volume. Again measure the volume and pressure. Let
pressure factor = ratio of pressure increase
volume factor = ratio of volume increase.

Gamma constant (γ)= Ln(pressure factor)/Ln(volume factor)

Beta constant (Bk) = Gamma - 1

How about Helium?

Last, Beta for Helium is about 0.7, meaning the same Work Energy, divided by the Heat Energy stored, is almost twice that of air. Which is why Helium behaves so much differently than air, in situations affected by molecular speed. At a given temperature, the Helium stores 4/7ths as much energy as air .Conversely Helium molecules travel much faster than air molecules, at the same temperature. Speed of sound is limited by the speed of gas molecules, hence is faster in Helium than air. (Yes, squeaky chipmunk voices are because sound travels faster in Helium -- because the molecules fly faster. Makes resonance in your voice box a higher pitch, because the sound waves bounce back sooner. )

Point being, Helium is VERY different from air, which is obvious by comparing Beta of 0.7 to Beta 0.4. The relative difference between their Gamma of 1.7 and 1.4 is misleadingly small. Why? Gamma is primarily a function of geometry, gas density halves, as does pressure, if volume doubles. Beta is a pure ratio with specific heat, a property of the substance.

And no, speed of sound has no relation to work or compression heating. Work (or heating) is the same whether volume change is fast or slow. This may appear to happen in a cooled engine, because there is a race with the speed the air is being cooled, which is a function of time.