Question

Instead of assuming that the volume of a condensed phase is constant when pressure is applied, assume only that the compressibility is constant.

1. Show that when the pressure is changed isothermally by ΔP, G changes to

G'=G+VmΔP(1-12κTΔP)

2. Assess the error in assuming that a solid is incompressible by applying this expression to the compression of copper when ΔP = 500 atm. (For copper at 25℃, κT = 0.8*10^-6 atm^-1 and ρ = 8.93 g/cm³.)

Answer 1

Ans. Given condition is volume of condensed phase is not constant, the change in volume of condensed phase will change the Gibbs free energy on application of pressure.

Considering actual Gibbs free energy = G (Joule / gmole)

When the given sample will have the variation in Gibbs free energy due to variation in volume of condensed phase. The additional Gibbs free energy parameter will be given as:

..................................(i)

Here, The additional change in Gibb's free energy are due to the variation in volume of the condensed phase.

Here V_m = volume of a given mass changed on application of pressure, in cm³.

amount of pressure applied during condensation process in atmosphere (atm).

compresibilty factor relation of a material with the variation in applied pressure during condensation (It is a unitless quantity)

Thus, the increase in Gibb's free energy due to variation in volume in condensation process is given as, the sum of initial Gibb's free energy to the additional Gibbs free energy due to applied pressure.

;.......................................(a)

Here, G' = Resulting Gibbs free energy (J/gmole)

Actual Gibbs free energy = G (Joule / gmole)

= Change in Gibbs free energy, due to variation in volume on application of pressure.

And   The additional change in Gibb's free energy (J/gmole)

The error in correction of temperature for the given sample is given as: = ,...(ii)

Here k_T = Thermal correction in material properties due to variation in volume, with the application of pressure,and is dependent on the amount of pressure applied. it is expressed in per atmosphere (atm^-1).

amount of pressure applied during condensation process in atmosphere (atm).

Thus, substituting (ii) in (i), we get:

......................... (iii)

= Change in Gibbs free energy, due to variation in volume on application of pressure.

.....................................................(iv)

amount of pressure applied during condensation process in atmosphere (atm).

Here V_m = volume of a given mass changed on application of pressure, in cm³.

Using relation (iii) and relation (iv), in relation (a) we get:

Hence proved.

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b). Here in the given relation of variation in Gibbs free energy,

,

The error will be given as:

Given: for copper at 25 ^oC,

,

material correction factor, k_T = 0.8 x10^-6 per atm,for copper at 25 ^oC,

Density of copper = 8.93 g/cm³, (Given).

Thus, volume (in cm³) of 1 gm of sample of copper is given as= 1/(Density of copper as 8.93 g/cm³)

The error for the pressure of 500 atm at 25 ^oC is calculated out as 55.7 J/gmole.

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