Integrate the expression β = 1/V (δV / δT) at constant pressure, assuming that β is independent of temperature. By doing so, obtain an expression for V as a function of T and β at constant P.
To solve this problem, we will use the integral calculus and the given equation:
β = 1/V (dV / dT) (Equation 1)
Our goal is to find an expression for V as a function of T and β at constant P. We will start by rearranging Equation 1 and separating variables:
βV = dV / dT
Next, we will integrate both sides of the equation:
∫ βV dT = ∫ dV
Integrating the left side with respect to T:
β ∫ V dT = ∫ dV
β∫ V dT = V + C
Where C is the constant of integration.
To find the expression for V, we'll integrate both sides once more, this time with respect to V:
β∫ dT = ∫ (V + C) dV
βT = ½ V^2 + CV + D
Where D is another constant of integration.
Since we are interested in V as a function of T and β, we rearrange the equation:
V^2 + 2CV + 2βT - 2D = 0
To determine the values of C and D, we can use the condition of constant pressure (P):
P = 2C + 2βT - 2D
Therefore, we can express C as a function of P, T, and β:
C = (P - 2βT + 2D) / 2
By substituting this value back into the equation, we get:
V^2 + (P - 2βT + 2D)V + 2βT - 2D = 0
Now, let's simplify the equation by grouping the terms:
V^2 + (P - 2βT)V + (2βT - 2D) = 0
Finally, we express V as a function of T and β at constant P:
V(T, β) = (- (P - 2βT) ± √((P - 2βT)^2 - 4(2βT - 2D))) / 2
This is the desired expression for V as a function of T and β at constant P.