This result is known as the reciprocal rule of partial differentiation.
This result is also known as the cyclic rule of partial differentiation. Hence, deduce that
where is the molar ideal gas constant.
By analyzing one-dimensional compressions and rarefactions of the system of fluid contained in a slab of thickness , show that the pressure, , in the fluid depends on the position, , and the time, , so as to satisfy the wave equation
where the velocity of sound propagation, , is a constant given by . Here is the equilibrium mass density of the fluid, and is its adiabatic compressibility,
that is, its compressibility measured under conditions in which the fluid is thermally insulated.
where is the isothermal compressibility. Hence, deduce that
where is the molar volume, and the volume coefficient of expansion.
where is the ratio of specific heats, and the gas pressure.
where is the mass density, the molar ideal gas constant, the absolute temperature, and the molecular weight.
where is the volume coefficient of expansion, and the isothermal compressibility.
derive the other three by making use of the reciprocal and cyclic rules of partial differentiation (see Exercises 1 and 2), as well as the identity
The critical point is defined as the unique point at which . (See Section 9.10.) Let , , and be the temperature, molar volume, and temperature, respectively, at the critical point. Demonstrate that , , and . Hence, deduce that the van der Waals equation of state can be written in the reduced form
where , , and .
where is the compression ratio of the engine.