Integrate the expression α=1/V (∂V/∂T) P assuming that α is independent of pressure. By doing so, obtain an expression for V as a function of T and α at constant P.

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Mathematics

Omari Davila

16 August, 01:10

Integrate the expression α=1/V (∂V/∂T) P assuming that α is independent of pressure. By doing so, obtain an expression for V as a function of T and α at constant P.

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Spencer Ferguson · Answer 1

V (T) = Peᵅᵀ

Step-by-step explanation:

α = 1/V (dV/dT) P

Rearranging and separating variables

(α/P) dT = (1/V) dV

Taking the integrals of both sides

∫ (α/P) dT = ∫ (1/V) dV

αT/P = ln V + ln C, C a constant of integration

Taking the exponent of both sides

exp (αT/P) = exp (ln V + ln C)

exp (αT/P) = exp (ln V) X exp (ln C)

CV=exp (αT/P)

Since Pressure is constant, exp (P) = Constant, say K.

V (T) = Peᵅᵀ where Pressure = K/C