# Isentropic Process, General Results for dU and dH

• I
• Kushwoho44
In summary, the conversation is discussing the relationship between change in internal energy and work done in an isentropic scenario. For an ideal gas, internal energy is only dependent on temperature and the heat capacity at constant volume is defined as the partial derivative of internal energy with respect to temperature at constant volume.
Kushwoho44
TL;DR Summary
I'm having trouble understanding intuitively the relation between the LHS and RHS of
dU = n*c_v *dT = -pdV.
Hello forumites,

I've been working with the following expression for the change in internal energy in an isentropic scenario.
$$dU = n*c_v *dT = -pdV$$

However, I'm a bit stumped here, the left hand side of the expression (or middle rather), states the change in internal energy is the product of the specific heat for constant volume and temperature, but this is equal to the work done on the system, which is the product of pressure and the volume differential.

This is confusing to me. We first invoke a constant-volume argument and then on the right hand side, state that it's equal to an expression dependent on a change in volume.

Any help would as always be appreciated.

For an ideal gas, internal energy is a function only of temperature, and is independent of pressure and volume. We use the heat capacity at constant volume, because this parameter is defined precisely in terms of the partial derivative of internal energy with respect to temperature at constant volume:
$$c_v\equiv \frac{1}{n}\left(\frac{\partial U}{\partial T}\right)_v$$
For an ideal gas, this reduces to:$$c_v= \frac{1}{n}\frac{d U}{dT}$$

## 1. What is an isentropic process?

An isentropic process is a thermodynamic process in which the entropy of a system remains constant. This means that the energy transfer into or out of the system occurs without any heat being added or removed.

## 2. What are the general results for dU and dH in an isentropic process?

The general results for dU (change in internal energy) and dH (change in enthalpy) in an isentropic process are both equal to zero. This is because, in an isentropic process, there is no heat transfer, so the change in internal energy and enthalpy are solely determined by work done on or by the system.

## 3. How is an isentropic process different from an adiabatic process?

An isentropic process and an adiabatic process are often used interchangeably, but there is a subtle difference between the two. An isentropic process is a process in which the entropy remains constant, while an adiabatic process is a process in which no heat is transferred. In other words, an isentropic process can be adiabatic, but an adiabatic process may not necessarily be isentropic.

## 4. What is the equation for calculating the change in entropy in an isentropic process?

The equation for calculating the change in entropy in an isentropic process is dS = 0. This means that the change in entropy is equal to zero, indicating that the entropy remains constant throughout the process.

## 5. Can an isentropic process occur in a closed system?

Yes, an isentropic process can occur in a closed system. In a closed system, no mass is exchanged with the surroundings, but energy can still be transferred in the form of work or heat. As long as there is no heat transfer, the process can be considered isentropic.

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