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Homework answers / question archive / dP/dT = L/T deltaV, is a differential equation that can, in principle, be solved to find the shape of the entire phase-boundary curve
dP/dT = L/T deltaV,
is a differential equation that can, in principle, be solved to find the shape of the entire phase-boundary curve. To solve it, however, you have to know how both L and deltaV depend on temperature and pressure. Often, over a reasonably small section of the curve, you can take L to be constant. Moreover, if one of the phases is a gas, you can usually neglect the volume of the condensed phase and just take deltaV to be the volume of the gas, expressed in terms of temperature and pressure using the ideal gas law. Making all these assumptions, solve the differential equation explicitly to obtain the following formula for the phase boundary curve:
P = (constant) x e-L/RT (the vapor pressure equation).
dP/dT = L/{T* del(V)}
If del(V) = V, then,
dP/dT = [L/(T*V)] ----(1)
From ideal gas law, PV = RT
Hence, V = (RT)/P
Substitute this V in eq. (1) and we get,
dP/dT = (L/T)*(P/RT) = LP/(R*T^2)
Hence, dP/P = (L/R)*dT/(T^2)
Integrate both sides and we get,
ln P = (L/R)*(-1/T) + c, where c is a constant
Hence P = e^[-(L/RT)+c] = e^c * e^[-(L/RT)]
or, P = constant * e^[-L/RT)] QED
