# How Do You Derive β for the Given Partial Differential Equation?

• j-lee00
In summary, the conversation involves the calculation of the derivative of the equation of state, pV = RT(1+B(T)/V), to find the variable \beta. The final result is \beta = (RT(B+V+T*dB/dT))/(TV(pV+RTB/V)).
j-lee00
PV = RT(1+B(T)/V)

$$\beta$$ = (1/V)*($$\frac{dV}{dT}$$) at constant P

show $$\beta$$ =$$\frac{1}{T}$$*$$\frac{V + B + T\frac{dB}{dT}}{V + 2B}$$

I got to

$$\beta$$ =$$\frac{PV}{VRT+PRTB}$$*($$\frac{R}{P}$$+$$\frac{d}{dT}$$$$\frac{RTB}{V}$$)

I need help with $$\frac{d}{dT}$$$$\frac{RTB}{V}$$)

I don't know the latex format for pd

Last edited:
Assuming this is the equation of state (click on any formula in a post to see the latex coding):

$$pV=RT\left(1+\frac{B(T)}{V} \right)$$

The way to proceed is to calculate the derivative

$$\frac{\partial V}{\partial T}$$

By taking the derivative of the equation of state. This gives you:

$$p\frac{\partial V}{\partial T}= R\left(1+\frac{B(T)}{V}\right)+ RT\left(\frac{1}{V}\frac{\partial B(T)}{\partial T}- \frac{B(T)}{V^2}\frac{\partial V}{\partial T}\right)$$

Or:

$$\left[ p+\frac{RTB(T)}{V^2}\right] \frac{\partial V}{\partial T}= R+ \frac{RB(T)}{V}+ \frac{RT}{V} \frac{\partial B(T)}{\partial T}$$

Now the question was to calculate:

$$\beta=\frac{1}{V}\frac{\partial V}{\partial T}$$

Thus you have (after rewriting):

$$\beta=\frac{RT\left(B+V+T\frac{\displaystyle \partial B(T)}{\displaystyle \partial T}\right)}{TV \left(pV+\frac{\displaystyle RTB(T)}{\displaystyle V}\right)}$$

Using now the given equation:

$$\frac{pV}{RT}=1+\frac{B(T)}{V}$$

and substitute in the denominator, you obtain the result asked for.

Cheers

You're welcome.

## 1. What is a partial differential equation (PDE)?

A PDE is a mathematical equation that involves multiple independent variables and their partial derivatives. It is used to describe how a quantity changes over time and space. PDEs are widely used in physics, engineering, and other scientific fields to model complex systems.

## 2. How is a PDE different from an ordinary differential equation (ODE)?

The main difference between a PDE and an ODE is that a PDE involves multiple independent variables, while an ODE involves only one independent variable. This means that a PDE describes how a quantity changes in multiple dimensions, while an ODE describes how a quantity changes along a single dimension.

## 3. What are some common types of PDEs?

Some common types of PDEs include the heat equation, wave equation, Laplace's equation, and the diffusion equation. These equations are used to model various physical phenomena, such as heat flow, wave propagation, and diffusion processes.

## 4. How are PDEs solved?

PDEs can be solved using a variety of techniques, such as separation of variables, Fourier transforms, and numerical methods. The specific method used depends on the type of PDE and the boundary conditions of the problem.

## 5. What are some applications of PDEs?

PDEs have many applications in science and engineering, including fluid dynamics, electromagnetism, quantum mechanics, and financial mathematics. They are used to model and predict the behavior of complex systems and are essential tools in understanding the world around us.

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