# What Is An Isothermal Process? (examples, Exercises)

The isothermal or isothermal process 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 does in physical characteristics.

These changes are the 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 extracting 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 caloric 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 human beings when the body temperature rises or falls we feel sick, because in our body many chemical reactions that maintain life take place at a constant temperature. This is true for warm-blooded animals in general.

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

## Examples of isothermal processes

-The metabolism of warm-blooded animals is carried out at a constant temperature.

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

-Melting ice is another common isothermal process, as is placing water in the freezer to make ice cubes.

-Automotive engines, refrigerators, as well as many other types of machinery, operate correctly in a certain temperature range. Devices called thermostats are used to maintain the proper temperature . 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 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 are adiabatic. The isothermal stages are compression and expansion of a gas that is responsible for producing useful work.

A car 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 thermodynamic principles are common.

## Calculation of the 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 finally W , which is the work done by said system.

Suppose that the system in question is an ideal gas contained in the cylinder of a moving piston of area A , doing work when its volume V changes from V 1 to V 2 .

The equation of state of ideal gas is PV = nRT , which relates the volume with the pressure P and temperature T . The values ​​of n and R are constant: n is the number of moles of the gas and R is the constant of the gases. In the case of an isothermal process the PV product is constant.

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

Since Adx is the precisely the volume variation dV , then:

dW = PdV

To obtain the total work in an isothermal process, we integrate the expression for dW:

$W=\int_{V_{1}}^{V_{2}}\frac{nRT}{V}dV=nRT&space;\:&space;ln\frac{V_{2}}{V_{1}}$

Pressure P and volume V are plotted on a PV diagram like the one shown in the figure, and the work done is equal to the area under the curve:

Since ΔU = 0 since the temperature remains constant, in an isothermal process we have:

Q = W

### – Exercise 1

A cylinder fitted with a moving piston contains an ideal gas at 127ºC. If the piston moves to reduce the initial volume 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.

Data : R = 8.3 J / mol. K

#### Solution

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 previously deduced equation:

$W=nRT&space;\:&space;ln\frac{V_{2}}{V_{1}}$

127 º C = 127 + 273 K = 400 K

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 (V 2 / 10V 2 ) = 5 moles

Work was preceded by a negative sign. The attentive reader will have noticed in the preceding section that W was defined as “work done by the system” and has a + sign. So the “work done on the system” has a negative sign.

### – Exercise 2

You have air in a cylinder fitted with a plunger. Initially there is 0.4 m 3 of gas at 100 kPa pressure and 80ºC temperature. The air is compressed to 0.1 m 3 ensuring that the temperature inside the cylinder remains constant during the process.

Determine how much work is done during this process.

#### Solution

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

80 º C = 80 + 273 K = 353 K.

P 1 V 1 = nRT → n = P 1 V 1 / RT = 100000 Pa x 0.4 m 3 /8.3 J / mol. K x 353 K = 13.65 mol

W = nRT ln (V 2 / V 1 ) = 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 was done on the system, which always happens when gas is compressed.

## References

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