# Work done in adiabatic process vs work done in isothermal

• shihab-kol
In summary, the question asks which process, adiabatic or isothermal, will result in more work being done when an ideal gas is compressed to the same volume from the same initial state. After analyzing the equations for both processes, it is concluded that the adiabatic process will result in more work being done, as the slope of the adiabat is higher than that of the isothermal, resulting in a higher final pressure and thus more work.
shihab-kol

## Homework Statement

[/B]
An ideal gas is compressed to the same volume from the same initial state for both an adiabatic and an isothermal process. In which case will more work be done ?

2. Homework Equations

##dU=dQ - dW ##
##W=\int P\,dV ##(For isothermal)
##W=nc_vdT##
##W=\frac{(P_1V_1-P_2V_2)} {ϒ-1}##(For adiabatic)
##PV^ϒ=constant##(For adiabatic)

## The Attempt at a Solution

I take the initial state to be ##(P_1,V_1)## and the final as ##(P_2,V_2)## for isothermal and ##(P_3,V_2)## for adiabatic.
I have $$PV=nRT$$
Differentiating with respect to V,
(For isothermal)[/B]
$$\frac{dP_i}{dV_i}=-\frac{P}{V}$$
And,
(For adiabatic)$$\frac{d(PV^ϒ)}{dV}= 0$$
Or,$$\frac{dP_a}{dV_a}=-(ϒ-1)\frac{P}{V}$$
Substituting the earlier equation into the above,$$\frac{dP_a}{dV_a} = (ϒ-1)\frac{dP_i}{dV_i}$$
What I get is that the slope of an adiabat is higher than the slope of an isothermal having the same initial state and thus,
$$P_3>P_2$$
And,
$$W_{Adiabatic}>W_{Isothermal}$$
Is this alright ?

Your answer is correct.

shihab-kol
Chestermiller said:
Your answer is correct.
Thanks for taking the trouble of checking it out.

## 1. What is an adiabatic process?

An adiabatic process is a thermodynamic process in which there is no heat transfer between the system and its surroundings. This means that the system is thermally insulated and no energy is exchanged in the form of heat.

## 2. What is an isothermal process?

An isothermal process is a thermodynamic process in which the temperature of the system remains constant. This means that the system is in thermal equilibrium with its surroundings and there is no change in the internal energy of the system.

## 3. How is work done in an adiabatic process different from work done in an isothermal process?

In an adiabatic process, work is done solely by the change in internal energy of the system, as there is no heat transfer. In an isothermal process, work is done by both the change in internal energy and the heat transfer.

## 4. Which process, adiabatic or isothermal, is more efficient in performing work?

An adiabatic process is more efficient in performing work because all the energy transferred is used to do work, while in an isothermal process, some energy is lost as heat transfer.

## 5. Can you give an example of a real-life adiabatic and isothermal process?

A common example of an adiabatic process is the compression and expansion of air in a car's engine, as there is no heat exchange during the process. An isothermal process can be seen in a refrigerator, where the refrigerant is compressed and expanded at a constant temperature to remove heat from the interior.

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