By Sunil Bhardwaj

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Consider an open system composed of i constituents. Let $${ n }_{ 1 }, { n }_{ 2 }, { n }_{ 3 }, ...., { n }_{ i }$$ be the number of moles of constituents 1, 2, 3, ...., i respectively. Let X be the extensive property. Its value is determined by the state of the system (i.e. Temperature and Pressure) and the amount of various constituents in the system. It means property X is a function of temperature, pressure and amount of various constituents. $$X = f\left( T, P, { n }_{ 1 }, { n }_{ 2 }, { n }_{ 3 }, ...., { n }_{ i } \right) \qquad .... (1)$$ The change in property dx due to small change in temperature, pressure and amount of constituents is given by, $$dx = { \left( \frac { \partial X }{ \partial T } \right) }_{ P, { n }_{ 1 }, { n }_{ 2 }, { n }_{ 3 }, ...., { n }_{ i } }dp + { \left( \frac { \partial X }{ \partial P } \right) }_{ T, { n }_{ 1 }, { n }_{ 2 }, { n }_{ 3 }, ...., { n }_{ i } }dt + { \left( \frac { \partial X }{ \partial { n }_{ 1 } } \right) }_{ T, P, { n }_{ 2 }, { n }_{ 3 }, ...., { n }_{ i } }d{ n }_{ 1 } + { \left( \frac { \partial X }{ \partial { n }_{ 2 } } \right) }_{ T, P, { n }_{ 1 }, { n }_{ 3 }, ...., { n }_{ i } }d{ n }_{ 2 } + { \left( \frac { \partial X }{ \partial { n }_{ 3 } } \right) }_{ T, P, { n }_{ 1 }, { n }_{ 2 }, ...., { n }_{ i } }d{ n }_{ 3 } + .... + { \left( \frac { \partial X }{ \partial { n }_{ i } } \right) }_{ T, P, { n }_{ 1 }, { n }_{ 2 }, { n }_{ 3 }, ...., }d{ n }_{ i }$$ $$\therefore \boxed { dx = { \left( \frac { \partial X }{ \partial T } \right) }_{ P, { n }_{ 1 }, { n }_{ 2 }, { n }_{ 3 }, ...., { n }_{ i } }dp + { \left( \frac { \partial X }{ \partial P } \right) }_{ T, { n }_{ 1 }, { n }_{ 2 }, { n }_{ 3 }, ...., { n }_{ i } }dt + \overline { { X }_{ 1 } } d{ n }_{ 1 } + \overline { { X }_{ 2 } } d{ n }_{ 2 } + \overline { { X }_{ 3 } } d{ n }_{ 3 } + .... + \overline { { X }_{ i } } d{ n }_{ i } }$$ Where $$\overline { { X }_{ 1 } }$$ is the partial molal property and $$\overline { { X }_{ 1 } } = { \left( \frac { \partial X }{ \partial { n }_{ 1 } } \right) }_{ T, P, { n }_{ 2 }, { n }_{ 3 }, ...., { n }_{ i } }$$

And at constant Temperature and Pressure $$\boxed { dx = \overline { { X }_{ 1 } } d{ n }_{ 1 } + \overline { { X }_{ 2 } } d{ n }_{ 2 } + \overline { { X }_{ 3 } } d{ n }_{ 3 } + .... + \overline { { X }_{ i } } d{ n }_{ i } }$$ In other words, the partial molal property is the change in the property X of system at constant temperature and pressue when one mole of a particular constituent is added to such a large quantity of the system that the added mole does not effect the composition of the system.

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