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.

∫ β δT = ∫ 1/V δV

Since β independant of T, ∫ β δT = βT + constant.
βT = ∫ 1/V δV

Use ∫ 1/x dx = ln(x) + constant
βT = ln(V) + constant

.: V = k e^(βT)

Thanks!!

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

Starting with the given equation:

β = 1/V (δV / δT)

Rearranging, we have:

V * β = δV / δT

To integrate, we can rewrite the equation in differential form:

V * β dT = dV

Now we can integrate both sides of the equation.

∫V * β dT = ∫dV

Integrating the left side with respect to T gives:

∫V * β dT = ∫dV

V * ∫β dT = ∫dV

V * β * T = V + C

Where C is the constant of integration.

Next, we can solve for V in terms of T and β at constant pressure (P).

Rearranging the equation, we get:

V * β * T - V = C

Factoring out V, we have:

V (β * T - 1) = C

Finally, we can express V in terms of T and β at constant pressure:

V = C / (β * T - 1)

That's the expression for V as a function of T and β at constant pressure (P).

To integrate the given expression β = 1/V (δV / δT) at constant pressure, we can rewrite it as:

δV / δT = βV

Notice that δV / δT represents the derivative of V with respect to T at constant pressure (holding pressure constant), which we can denote as dV / dT. Also, since β is independent of temperature, we can treat it as a constant.

So, the equation becomes:

dV / dT = βV

To solve this separable differential equation, we can rewrite it as:

dV / V = βdT

Now, we'll integrate both sides of the equation:

∫(dV / V) = ∫βdT

Integrating the left side gives us ln|V|, and integrating the right side gives us βT + C, where C is the constant of integration. Thus, we have:

ln|V| = βT + C

To eliminate the absolute value, we can rewrite it as:

|V| = e^(βT+C)

Considering that C is an arbitrary constant, we can combine it with another constant, say A. Thus, we have:

|V| = Ae^(βT)

Finally, we can remove the absolute value by making V positive, so:

V = Ae^(βT)

Therefore, the expression for V as a function of T and β at constant P is V = Ae^(βT), where A is a constant determined by initial conditions or any other constraints specific to the problem.