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The speed of sound in a gas at temperature T is given to be ## v=\sqrt{\frac{\gamma RT}{M}}##, where ##\gamma## is the adiabatic exponent, R is the gas constant and M is the molar mass of the gas. In deriving this expression, we assumed that the compression and expansion processes were so fast that there isn't enough time for heat transfer to take place, and therefore that the processes are approximately adiabatic (Laplace's correction to Newton's formula). We therefore used ##PV^\gamma = constant##. However the relation ##PV^\gamma = constant## is valid only for a quasi-static adiabatic process, and cannot be used for very fast processes. The quasi-static assumption is made in the derivation of adiabatic process relations. So how are we allowed to use it in the speed of sound derivation? The very reason we assumed the process was adiabatic is because it is very fast in the first place. So how is it consistent to assume it is quasi-static as well?

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