8 03 at ESG Supplemental Notes Adiabatic Bulk Moduli To find the speed of sound in a gas or any property of a gas involving elasticity see the discussion of the Helmholtz oscillator B B page 22 or B B problem 1 6 or French Pages 57 59 we need the bulk modulus of the fluid This will correspond to the spring constant of a spring and will give the magnitude of the restoring agency pressure for a gas force for a spring in terms of the change in physical dimension volume for a gas length for a spring It turns out to be more useful to use an intensive quantity for the bulk modulus of a gas so what we want is the change in pressure per fractional change in volume so the bulk modulus denoted as the Greek kappa which when written has a great tendency to look like k and in fact French uses K is p V V or V dp dV The minus sign indicates that for normal fluids not all are a negative change in volume results in an increase in pressure To find the bulk modulus we need to know something about the gas and how it behaves For our purposes we will need three basic principles actually 2 1 2 which we get from thermodynamics or the kinetic theory of gasses You might well have encountered these in previous classes such as chemistry A The ideal gas law pV nRT with the standard terminology B The first law of thermodynamics dU dQ p dV where U is internal energy and Q is heat C a consequence of B so call it B 1 2 The internal energy of a gas is a state variable and hence is a function of temperature This allows us to say that dU n CV dT where CV is the molar heat capacity at constant volume What this means is that any process that changes U will change U by the same amount as a constant volume process with the same temperature change would Specifically for a constant pressure process dU n Cp dT p dV where Cp is the molar heat capacity at constant pressure So we have two expressions for dU and they must be the same n CV dT n Cp dT p dV 1 Using the ideal gas law for a constant pressure process dp 0 n R dT p dV so the above relation between CV and Cp becomes n CV dT n Cp dT n R dT or Cp CV R From this we se that Cp CV This is what is expected from physical grounds if a gas is free to expand constant pressure it takes more heat to increase the temperature than if the gas is confined It will be convenient to define the quantity CCVp 1 For an adiabatic process no heat is added and dQ 0 Note that this does not mean that dT 0 in fact for an adiabatic process none of the intensive quantities p T or the molar volume V n will be constant Equating expressions for dU dU n CV dT p dV from p V n R T n R dT V dp p dV so upon substitution of dT rearrangement and cancellation of the factor n CV V dp p dV R p dV CV V dp R CV p dV V dp p dV At this point we can get our result for the adiabatic bulk modulus V dp p dV We can also get a more general result relating the pressure and volume for an adiabatic process p V constant which is useful when a calculation of work the integral of p dV is needed In practice isothermal dT 0 processes for gasses are rather rare adding heat to a system in such a way that the temperature remains uniform is difficult and is often just an idealization For such a process however the isothermal bulk modulus is simply p Note that this corresponds to 1 or a system in which the heat capacities at constant volume and constant pressure are nearly identical This is often the case for condensed matter but not for gasses 2