8 julio, 2024

What is an isothermal process? (Examples, exercises)

He isothermal process or isothermal is a reversible thermodynamic process in which the temperature remains constant. In a gas, there are situations in which a change in the system does not produce variations in temperature, but in the physical characteristics.

These changes are phase changes, when the substance changes from solid to liquid, from liquid to gas, or vice versa. In such cases, the molecules of the substance readjust their position, adding or removing thermal energy.

The thermal energy required for a phase change to occur in a substance is called latent heat or heat of transformation.

One way to make a process isothermal is to put the substance that will be the system under study in contact with an external thermal reservoir, which is another system with a large heat capacity. In this way, such a slow heat exchange occurs that the temperature remains constant.

This type of process occurs frequently in nature. For example, in humans when the body temperature rises or falls we feel sick, because in our body many chemical reactions that maintain life occur at constant temperature. This is true for warm-blooded animals in general.

Other examples are the ice that melts in the heat when spring arrives and the ice cubes that cool the drink.


Examples of isothermal processes

-The metabolism of warm-blooded animals takes place at constant temperature.

-When water boils, a phase change occurs, from liquid to gas, and the temperature remains constant at approximately 100ºC, since other factors can influence the value.

-Ice melting is another frequent isothermal process, as is placing water in the freezer to make ice cubes.

-Automobile engines, refrigerators, as well as many other types of machinery, operate correctly in a certain temperature range. Devices called thermostats. Various operating principles are used in its design.

The Carnot cycle

A Carnot engine is an ideal machine from which work is obtained thanks to entirely reversible processes. It is an ideal machine because it does not consider processes that dissipate energy, such as the viscosity of the substance that does the work, nor friction.

The Carnot cycle consists of four stages, two of which are precisely isothermal and the other two adiabatic. The isothermal stages are compression and expansion of a gas that is responsible for producing useful work.

An automobile engine operates on similar principles. The movement of a piston inside the cylinder is transmitted to other parts of the car and produces movement. It does not have the behavior of an ideal system like the Carnot engine, but the thermodynamic principles are common.

Calculation of work done in an isothermal process

To calculate the work done by a system when the temperature is constant, we must use the first law of thermodynamics, which states:

ΔU = Q – W

This is another way of expressing the conservation of energy in the system, presented through ΔU or change in energy, Q as the heat supplied and lastly Wwhich is the work done by the system.

Suppose that the system in question is an ideal gas contained in the cylinder of a moving piston of area TOwhich does work when its volume V change of V1 to V2.

The ideal gas equation of state is PV=nRTwhich relates volume to pressure P and the temperature you. The values ​​of n and R are constants: n is the number of moles of the gas and R is the gas constant. In the case of an isothermal process, the product PV it’s constant.

Well then, the work done is calculated by integrating a small differential work, in which a force F produces a small displacement dx:

dW= Fdx = PAdx

As adx is the precisely the change in volume dVso:


To obtain the total work in an isothermal process, the expression for dW is integrated:

The pressure P and the volume V are plotted on a diagram PV as shown in the figure and the work done equals the area under the curve:

As ΔU = 0 Since the temperature remains constant, in an isothermal process one has to:


– Exercise 1

A cylinder fitted with a movable piston contains an ideal gas at 127°C. If the piston is moved until the initial volume is reduced to 10 times, keeping the temperature constant, find the number of moles of gas contained in the cylinder, if the work done on the gas is 38.180 J.

Fact: R = 8.3 J/mol. k


The statement states that the temperature remains constant, therefore we are in the presence of an isothermal process. For the work done on the gas we have the equation previously deduced:

The initial volume V1 is 10 times the final volume V2, therefore V1 = 10V2. Very important: Before substituting the data, note that the gas constant has units of the International System, therefore it is necessary to convert degrees Celsius to kelvin:

127ºC = 127 + 273K = 400K

Solve for n, the number of moles:

n = W / RT ln(V2 / V1) = -38180 J / 8.3 J/mol.K x 400 K x ln (V2/10V2) = 5 moles

Work was placed before a negative sign. The attentive reader will have noticed in the preceding section that W was defined as «the work done by the system» and carries a + sign. Then the «work done on the system» has a negative sign.

– Exercise 2

Air is in a cylinder fitted with a piston. Initially there is 0.4 m3 of gas at a pressure of 100 kPa and a temperature of 80ºC. The air is compressed to 0.1 m3 making sure that the temperature inside the cylinder remains constant during the process.

Determine how much work is done during this process.


We use the equation for work derived previously, but the number of moles is unknown, which can be calculated from the ideal gas equation:

80ºC = 80 + 273K = 353K.

P1V1 = nRT → n = P1V1 /RT = 100000 Pa x 0.4 m3 /8.3 J/mol. K x 353 K = 13.65 mol

W = nRT ln(V2/V1) = 13.65 mol x 8.3 J/mol. K x 353 K x ln (0.1 /0.4) = -55,442.26 J

Again the negative sign indicates that work has been done on the system, which always happens when gas is compressed.


Bauer, W. 2011. Physics for Engineering and Science. Volume 1. Mc Graw Hill. Cengel, Y. 2012. Thermodynamics. 7th Edition. McGraw Hill. Figueroa, D. (2005). Series: Physics for Science and Engineering. Volume 4. Fluids and Thermodynamics. Edited by Douglas Figueroa (USB). Knight, R. 2017. Physics for Scientists and Engineering: a Strategy Approach. Serway, R., Vulle, C. 2011. Fundamentals of Physics. 9th Cengage Learning. Wikipedia. Isothermal Process. Retrieved from: en.wikipedia.org.

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