Solution Found!
A formula analogous to that for CP – CV relates the
Chapter 5, Problem 16P(choose chapter or problem)
A formula analogous to that for CP – CV relates the isothermal and isentropic compressibilities of a material:
(Here κS = – (1/V)(∂V/∂P)S is the reciprocal of the adiabatic bulk modulus considered in Problem.) Derive this formula. Also check that it is true for an ideal gas.
Problem:
By applying Newton’s laws to the oscillations of a continuous medium, one can show that the speed of a sound wave is given by
where ρ is the density of the medium (mass per unit volume) and B is the bulk modulus, a measure of the medium’s stiffness. More precisely, if we imagine applying an increase in pressure ΔP to a chunk of the material, and this increase results in a (negative) change in volume ΔV, then B is defined as the change in pressure divided by the magnitude of the fractional change in volume:
This definition is still ambiguous, however, because I haven’t said whether the compression is to take place isothermally or adiabatically (or in some other way).
(a) Compute the bulk modulus of an ideal gas, in terms of its pressure P, for both isothermal and adiabatic compressions.
(b) Argue that for purposes of computing the speed of a sound wave, the adiabatic B is the one we should use.
(c) Derive an expression for the speed of sound in an ideal gas, in terms of its temperature and average molecular mass. Compare your result to the formula for the rms speed of the molecules in the gas. Evaluate the speed of sound numerically for air at room temperature.
(d) When Scotland’s Battlefield Band played in Utah, one musician remarked that the high altitude threw their bagpipes out of tune. Would you expect altitude to affect the speed of sound (and hence the frequencies of the standing waves in the pipes)? If so, in which direction? If not, why not?
Questions & Answers
QUESTION: Problem 16P
A formula analogous to that for CP – CV relates the isothermal and isentropic compressibilities of a material:
(Here κS = – (1/V)(∂V/∂P)S is the reciprocal of the adiabatic bulk modulus considered in Problem.) Derive this formula. Also check that it is true for an ideal gas.
Problem:
By applying Newton’s laws to the oscillations of a continuous medium, one can show that the speed of a sound wave is given by
where ρ is the density of the medium (mass per unit volume) and B is the bulk modulus, a measure of the medium’s stiffness. More precisely, if we imagine applying an increase in pressure ΔP to a chunk of the material, and this increase results in a (negative) change in volume ΔV, then B is defined as the change in pressure divided by the magnitude of the fractional change in volume:
This definition is still ambiguous, however, because I haven’t said whether the compression is to take place isothermally or adiabatically (or in some other way).
(a) Compute the bulk modulus of an ideal gas, in terms of its pressure P, for both isothermal and adiabatic compressions.
(b) Argue that for purposes of computing the speed of a sound wave, the adiabatic B is the one we should use.
(c) Derive an expression for the speed of sound in an ideal gas, in terms of its temperature and average molecular mass. Compare your result to the formula for the rms speed of the molecules in the gas. Evaluate the speed of sound numerically for air at room temperature.
(d) When Scotland’s Battlefield Band played in Utah, one musician remarked that the high altitude threw their bagpipes out of tune. Would you expect altitude to affect the speed of sound (and hence the frequencies of the standing waves in the pipes)? If so, in which direction? If not, why not?
ANSWER:
Step 1 of 7
The isothermal compressibility has been defined as:
The adiabatic compressibility has been defined as:
The volume changes because of the temperature and the pressure changes.
Similarly, the pressure changes and the entropy changes also account for temperature change
From the equation 
From the equation 
