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Show that the differential d { V } of the molar volume of an i

Question

Show that the differential $d \overline{V}$ of the molar volume of an ideal gas is an exact differential and hence $\overline{V}$ is a state function.

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Asher Discussed

Show that the differential

d \overline{V}

of the molar volume of an ideal gas is an exact differential and hence

\overline{V}

is a state function.

9 mins ago

Discuss this question LIVE

9 mins ago

Text solutionVerified

\overline{V} = \frac{RT}{P}

For

d \overline{V}

to be exact differntial,

\frac{δ}{δ T} (\frac{δ V}{δ P}) = \frac{δ}{δ P} (\frac{δ V}{δ T})

\frac{δ}{δ T} (\frac{δ V}{δ P}) = \frac{δ}{δ T} (\frac{- RT}{p ^{2}}) = \frac{- R}{P ^{2}}

\frac{δ}{δ T} (\frac{δ V}{δ T}) = \frac{δ}{δ P} (\frac{R}{P}) = \frac{- R}{P ^{2}}

\frac{δ}{δ T} (\frac{δ V}{δ P}) = \frac{δ}{δ P} (\frac{δ V}{δ T}) = \frac{- R}{P ^{2}}

So,

d \overline{V}

is an exact differential.

Volume is a state function because the volume is only dependent on the final and initial values and not on the path taken to establish those values.

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Question Text	Show that the differential $d \overline{V}$ of the molar volume of an ideal gas is an exact differential and hence $\overline{V}$ is a state function.
Answer Type	Text solution:1
Upvotes	150