Question

1. Consider the volume of a substance as a function of pressure and temperature. Use the Euler chain relation and the definitions of the thermal expansivity and the isothermal compressibility to show that др ) - a LOT J KT is true. 2. At 363 K and 1 bar, liquid toluene has a thermal expansivity of 1.284 x 10'K' and an isothermal compressibility of 1.972 x 10 bar. Suppose a rigid container is completely filled with liquid toluene at 362.5 K and then the container is sealed. Suppose the temperature of the system increased to 363.5 K. By how much would the pressure within the container increase? (Assume a and Ky are constant over the temperature and pressure range involved.)

Answer 1

Solution:

Isothermal compressibility (k_T) = - 1/V ($\partial$V/$\partial$P)_T

Isothermal expansivity (α) = 1/V ($\partial$V/$\partial$T)p

Therefore,

α / kT = - ( $\partial$ V/$\partial$T)p / ($\partial$V/$\partial$P)_T

= ($\partial$T/$\partial$V)p / ($\partial$V/$\partial$P)_T -----(1)

Since, V = f (T, P)

Therefore, according to Euler reciprocity relation,

($\partial$T/$\partial$V)p . ($\partial$V/$\partial$P)_T. ($\partial$P/$\partial$T)v = -1 -----(2)

Thus, from equation 1 and 2,

α / k_T= ($\partial$T/$\partial$P)v