Isothermal Process - Definition, Example, Formula, FAQs

An isothermal process is a thermodynamic process in which the temperature of a system remains constant throughout. During this process, the system exchanges heat with its surroundings to ensure the temperature stays steady, despite changes in pressure or volume. One common example of an isothermal process is the slow compression or expansion of gas in a piston that is kept in thermal equilibrium with its environment. The internal energy of an ideal gas in such a system remains unchanged, as it depends solely on temperature.

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This Story also Contains

1. What is an Isothermal Process?
2. Solved Examples Based on Isothermal Process
3. Summary

In real life, we can observe isothermal processes in refrigeration and air conditioning systems, where gases are compressed and expanded while maintaining a constant temperature to regulate cooling. Similarly, natural processes, such as the slow melting of glaciers, can exhibit near-isothermal conditions, where temperature remains constant as the ice absorbs heat to transition into water without a significant rise in temperature.

What is an Isothermal Process?

An isothermal process is a type of thermodynamic process where the temperature of the system remains constant throughout the entire process. In this process, the system exchanges heat with its surroundings to maintain this constant temperature, despite changes in other properties like pressure or volume.

Some common isothermal process examples are listed as:

1. When ice melts at a temperature of zero degrees then the whole ice melts but the temperature of the system remains the same over the period of time, hence this is an example of an isothermal process.
2. All the thermodynamic reactions which occur inside the refrigerator are isothermal processes because the temperature of the refrigerator remains the same.
3. and, all such thermodynamic processes which are carried out at constant magnitude of temperature will be considered as isothermal processes.

What is Isotherm?

If we observe the relation between temperature and any other thermodynamic variable such as pressure, volume and others and draw a graph on the Cartesian plane then, all the curves which represent two states of a system where the temperature is the same during an isothermal process are called isotherms.

So, Those curves or lines which represent two states of a system at the same temperature in an isothermal process are called an isotherm.

For example, a line drawn as shown in the diagram below has two states A and B and both are at the same temperature during an isothermal process so this line is called an isotherm. We can consider any other thermodynamic variable on the X-axis.

Isothermal Process Formula

The basic formula in thermodynamics which shows that two states are in the isothermal process is simply written as P₁V₁=P₂V₂ where P, V represents the pressure and volume of an isothermal process in two states 1 and 2 and this is the Isothermal process formula.

Work Done in the Isothermal Process

When a system undergoes an isothermal process, either work is done on it or work is done by it and this work is different for different processes. In the isothermal process work done by the system is calculated using the formula. W=2.303RT log₁₀(V₂/V₁) where V represents volume at two different states being at a constant temperature of T and R is the universal gas constant.

So, work done in the isothermal process is W=2.303RT log₁₀(V₂/V₁)

PV Diagram for the Isothermal Process

A diagram representation of the pressure and volume of an isothermal process on a cartesian plane is called a PV diagram for an isothermal process and it is best shown in the diagram given below:

Change in Internal Energy in the Isothermal Process

The internal energy of any thermodynamic system is calculated as ∆U=nCv∆T where n represents the total number of moles of a gas, C represents the specific heat of the gas but at constant volume and T represents temperature. In an isothermal process, the change in temperature is zero because of the constant temperature in the isothermal process. So, In an isothermal process, the change in internal energy is zero.

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Solved Examples Based on Isothermal Process

Example 1: An ideal gas enclosed in a vertical cylindrical container supports a freely moving piston of mass M. The piston and the cylinder have cross-sectional area A. When the piston is in equilibrium, the volume of the gas is V₀ and its pressure is P₀ . The piston is slightly displaced from the equilibrium position and released. Assuming that the system is completely isolated from its surroundings, the piston executes a simple harmonic motion with frequency:

1) 12πMV0AγP0
2) 12πAγP0V0M
3) 12πV0MP0A2γ
4) 12πA2γP0MV0

Solution:

Relation between the slope of the isothermal and adiabatic process -
dPdVisothermal =−PVdPdVadiabatic =−γPV

wherein
Slope of adiabatic process =γ× Slope of isothermal process
At equilibrium

V=V0,P=P0
Let the atmospheric pressure be pa

(P0−Pa)A=Mg
Let the piston be displaced slightly into the cylinder then pressure becomes P0+dP

⇒ Net force =Mg−(P0+dP)A+PaAF=−dPA………..(2)

Since the cylinder is isolated, then the process is adiabatic ⇒PVr= constant
⇒Vγ+PγVγ−1dV=0 or dP=−γPVdV∴F=−γP0V0AdVdV=Adx Acceleration =F/M==−γA2P0MV0dx
This equation is of SHM with

w2=(γP0A2MV0)T=2πMV0γP0A2=1/T=12πγP0A2MV0

Hence, the answer is the option 4.

Example 2: The slope of isothermal and adiabatic curves are related as:

1) Isothermal curve slope = adiabatic curve slope
2) Slope of isothermal process =γ× Slope of adiabatic process
3) Slope of adiabatic process =γ× Slope of isothermal process
4) Slope of adiabatic process =12γ× Slope of isothermal process

Solution:

Relation between the slope of the isothermal and adiabatic process -
dPdVisothermal =−PVdPdVadiabatic =−γPV

wherein
Slope of adiabatic process =γ× Slope of isothermal process
For the isothermal process PV= constant

dPdV=−PV= slope of isothermal curve
For adiabatic process PVγ= constant

dPdV=−γPV= slope of adiabatic process so (dPdV)adiabatic =γ(dPdV)isothermal

Hence, the answer is the option 3.

Example 3: In the given diagram graph 1 represents the adiabatic process and 2 represents the isothermal process. Which of the following comparisons is not true for graphs 1 and 2

1) W1>W2
2) P1>P2
3) T1<T2
4) ( slope )1>( slope )2

Solution:

Comparison between isothermal and adiabatic processes in compression -

wherein
ωadia >ωisothermal Padia>Pisothermal Tadia >Tisothermal
Work done in process 1 is more than process 2 as the area under curve 1 is more
As shown in the diagram P1 is greater than P2

VfP∝T so T1>T2
Now for constant slope of 1 is γ times slope of curve 2

Hence, the answer is the option 3.

Example 4: Which of the following statements is true for the given diagram

1) If 1 is isothermal then 2 must be adiabatic

2) If 2 is adiabatic then 1 must be isothermal

3) Both may be isothermal

1) only 3

2) only 1

3) both 1 and 3

4) both 2 and 3

Solution:

Comparison between isothermal and adiabatic processes in expansion

wherein

ωisothermal >ωadia Pisothermal >PadiaTi sothermal >Tadia

If 1 is isothermal then 2 can not be isothermal because both can't intersect each other in this case so 2 must be adiabatic

If 2 is adiabatic then 1 may be adiabatic also because of different values of γ for monoatomic, diatomic and triatomic gases.

Hence, the answer is the option 2.

Example 5: The given diagram shows four processes i.e., isochoric, isobaric, isothermal and adiabatic. The correct assignment of the processes, in the same order, is given by:

1) a d b c

2) d a c b

3) a d c b

4) d a b c

Solution:

Comparison between isothermal and adiabatic processes in expansion

wherein

Wisothermal >Wadia Pisothermal >Padia Tisothermal >Tadia

a-isobaric (constant pressure) b- isothermal c- adiabatic d- isochoric (constant volume) For the same 'V' Pisothermal >Padiabatic for isothermal PV= const p∝1V for adiabatic PVγ= const γ=CpCv>1P∝1Vγ

Hence, the answer is the option 4.

Summary

An isothermal process is a thermodynamic process where the temperature remains constant throughout, with pressure and volume changing accordingly. Examples include ice melting at a constant temperature and processes inside a refrigerator. The work done in an isothermal process can be calculated using the formula W=2.303RTlog10⁡(V2/V1) and the change in internal energy is zero since there is no change in temperature. PV diagrams help visualize the relationship between pressure and volume in such processes, and comparisons are often made with adiabatic processes.