- Consider a relatively small region of a homogeneous substance, containing
particles, that is in equilibrium with the remainder
of the substance. The region is characterized by a volume,
, and a temperature,
.
- Show that the properly normalized probability that the
volume lies between
and
, and the temperature lies between
and
, is

Here, the mean temperature, , is the same as that of the remainder of the substance, whereas the mean volume, , is such that when the pressure of the region matches that of the surrounding substance. - Hence, deduce that
- Show that for the case of a monatomic ideal gas

where is the pressure of the region. Furthermore, demonstrate that

- Show that the properly normalized probability that the
volume lies between
and
, and the temperature lies between
and
, is
- A substance of molecular weight
has its triple point at the absolute temperature
and pressure
.
At this point, the mass densities of the solid and liquid phases are
and
, respectively, while the
vapor phase can be approximated as a dilute ideal gas. If, at the triple point, the slope of the melting curve is
,
and that of the liquid vaporization curve is
, show that the slope of the sublimation curve can be written
- The vapor pressure of solid ammonia (in millimeters of mercury) is given by
, and that of liquid ammonia by
. Here,
is
in degrees kelvin.
- Deduce that the triple point of ammonia occurs at K.
- Show that, at the triple point, the latent heats of sublimation, vaporization, and melting of ammonia are , , and , respectively.

- Water boils when its vapor pressure is equal to that of the atmosphere. The boiling point of pure water at
ground level is
C. Moreover, the latent heat of vaporization at this temperature is
.
Show that the boiling point of water decreases approximately linearly with increasing altitude such that
- The
*relative humidity*of air is defined as the ratio of the partial pressure of water vapor to the equilibrium vapor pressure at the ambient temperature. The*dew point*is defined as the temperature at which the relative humidity becomes . Show that if the relative humidity of air at (absolute) temperature is then the dew point is given by - When a rising air mass in the atmosphere becomes saturated (i.e., attains
relative humidity), condensing water droplets give up energy, thereby
slowing the adiabatic cooling process.
- Use the first law of thermodynamics to show that, as condensation forms during adiabatic expansion, the temperature
of the air mass changes by
- Assuming that the air is always saturated during this process, show that
- Use the equation of hydrostatic equilibrium of the atmosphere,
*wet adiabatic lapse-rate*of the atmosphere: - At C, the vapor pressure of water is bar, and the molar latent heat of vaporization is . At C, the vapor pressure of water is bar, and the molar latent heat of vaporization is . What is the ratio of the wet adiabatic lapse-rate to the dry adiabatic lapse-rate at these two temperatures?

- Use the first law of thermodynamics to show that, as condensation forms during adiabatic expansion, the temperature
of the air mass changes by
- Consider a phase transition in a van der Waals fluid whose reduced equation of state is
- Demonstrate that the Maxwell construction implies that
- Eliminate
and
from the previous equations to obtain the transcendental equation
- Writing
and
, show that the previous equation reduces to
- By setting both sides of the previous equation equal to
, show that it can be solved parametrically to give

where - Furthermore, demonstrate that

where

Here, and denote the molar entropies of the gas and liquid phases, respectively. - Finally, by considering the limits
and
, show that

in the limit , and

in the limit . Here, is the molar latent heat of vaporization.

- Demonstrate that the Maxwell construction implies that