# Deriving an expression for internal energy.

• L-x
In summary, the conversation discusses how to calculate the change in internal energy of a simple system between two states using the equations dU = dQ - pdV and dU = TdS - PdV. The final solution involves manipulating the equations and taking into account reversible and non-reversible processes.
L-x

## Homework Statement

Show that the change in internal energy of a simple system between states (V1, T1)
and (V2, T2) is given by

$$∆U = \int^{T1}_{T2} C_v\ dT + \int^{V1}_{V2} T.\frac{\partial p}{\partial T}|_V - p \ dV$$

dU=dQ-pdV

## The Attempt at a Solution

As U is a function of state i wrote down $$dU =\frac{\partial U}{\partial T}|_V dT + \frac{\partial U}{\partial V}|_T dV$$

$$\frac{\partial U}{\partial T}|_V$$ is clearly just Cv but i can't get the other part into the correct form, my manipulation is just going around in circles.

Keep in mind that, in terms of its fundamental variables, a differential in internal energy is also given by:

$$dU = T dS - P dV$$

You can then write:

$$\left( {\frac{{\partial U}}{{\partial V}}} \right)_T = T\left( {\frac{{\partial S}}{{\partial V}}} \right)_T - P$$

Can you figure out what to do from there?

danago said:
Keep in mind that, in terms of its fundamental variables, a differential in internal energy is also given by:

$$dU = T dS - P dV$$

Doesn't this only hold for a reversible process?

The equation is derived for a reversible process, however internal energy is a state function so it can be applied to non-reversible processes.

Ah of course! Thanks very much for your help.

## 1. What is internal energy?

Internal energy is the total energy contained within a system, including all the microscopic forms of energy such as the kinetic energy of particles, potential energy from intermolecular forces, and the energy associated with the system's overall temperature and pressure.

## 2. Why is it important to derive an expression for internal energy?

Deriving an expression for internal energy allows us to better understand the behavior and properties of a system. It also allows us to make predictions and calculations related to the system's energy and thermodynamic processes.

## 3. What factors are considered when deriving an expression for internal energy?

The factors that are typically considered when deriving an expression for internal energy include the number of particles in the system, the types of particles and their masses, the temperature and pressure of the system, and any external work or heat added or removed from the system.

## 4. How do you derive an expression for internal energy?

An expression for internal energy is typically derived using the laws of thermodynamics, specifically the first and second laws. This involves considering the different forms of energy within a system and how they change in response to changes in temperature, pressure, and other factors.

## 5. Can the expression for internal energy be applied to all systems?

The expression for internal energy can be applied to many different types of systems, including gases, liquids, and solids. However, it may need to be modified or adjusted for certain unique systems, such as systems with phase changes or chemical reactions.

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