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.

Question

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.

Answer 1

To integrate the expression β = 1/V (δV / δT) at constant pressure, we can rearrange the equation and separate the variables.

Start by rearranging the equation as follows:

β = (1/V) (δV / δT)

Multiply both sides by V and multiply both sides by δT to separate variables:

β * V * δT = δV

Now, we can integrate both sides of the equation.

Integrating the left side:

∫ β * V * δT = ∫ δV

β * ∫ V * δT = ∫ δV

β * ∫ V dT = ∫ δV

Since β is independent of temperature, it can be taken out of the integral on the left side:

β * ∫ V dT = ∫ δV

Next, integrate both sides to get the final result:

β * ∫ V dT = ∫ δV

β * ∫ V dT = V + C

∫ V dT = (V + C)/β

Integrating the right side results in the expression of V as a function of T and β at constant pressure:

∫ V dT = (V + C)/β

V * T = (V + C)/β

V * T * β = V + C

Now, rearranging the equation to isolate V:

V * (T * β - 1) = C

V = C / (T * β - 1)

Thus, we have obtained an expression for V as a function of T and β at constant pressure, which is V = C / (T * β - 1).

Answer 2

To integrate the expression β = 1/V (δV / δT) at constant pressure, we can rearrange it to solve for δV:

δV = β * V * δT

Next, we want to integrate this equation. Since the expression for β states that it is independent of temperature, we can treat β as a constant during the integration.

Integrating δV = β * V * δT, we get:

∫ δV = β * ∫ V * δT

The integral on the left side becomes the full V, and the integral on the right side becomes βT + C, where C is the constant of integration.

So, we have:

V = βT + C

Here, the constant C can be determined by using the condition that V is constant at a given constant pressure (P). At constant pressure, the volume V does not change, meaning that the expression βT + C should also be constant.

Therefore, we can write:

V = βT + C = constant

Since V is constant for a given pressure, C must be equal to -βT.

Substituting this value of C into the equation, we obtain:

V = βT - βT

Simplifying the expression, we get the final result:

V = βT(1 - 1) = 0

Hence, the expression for V as a function of T and β at constant pressure is V = 0.