### Overview

Source: Ketron Mitchell-Wynne, PhD, Asantha Cooray, PhD, Department of Physics & Astronomy, School of Physical Sciences, University of California, Irvine, CA

The second law of thermodynamics is a fundamental law of nature. It states that the entropy of a system always increases over time or remains constant in ideal cases when a system is in a steady state or undergoing a "reversible process." If the system is undergoing an irreversible process, the entropy of the system will always increase. This means that the change in entropy, ΔS, is always greater than or equal to zero. The entropy of a system is a measure of the number of microscopic configurations the system can attain. For example, gas in a container with known volume, pressure, and temperature can have an enormous number of possible configurations of the individual gas molecules. If the container is opened, the gas molecules escape and the number of configurations increases dramatically, essentially approaching infinity. When the container is opened, the entropy is said to increase. Therefore, entropy can be considered a measure of the "disorder" of a system.

### Principles

Entropy is a "state property," which is a quantity that depends only upon the current state of the system. Quantities that are state properties do not depend on the path by which the system arrived at its present state. Therefore, the most useful way to quantify a state property is to measure its *change*.

The change in entropy *S* is defined as:

*ΔS = Q / T, ***(Equation 1)**

*where Q* is the heat supplied to the system and *T* is the temperature of the system. In the context of thermodynamics, heat, like work, is defined as a transfer of energy. Heat is energy transferred from one object to another because of a difference in temperature. Consider a bath of ice and water at 0 °C. If one supplies heat to the ice/water bath, some of the ice will melt, and the number of states available to the water molecules will increase by a large amount, proportional to the amount of heat that was added to the system. The entropy will then increase proportionally to this amount. The relationship between two objects at different temperatures was first described by Newton.

Newton's law of cooling states that the rate of change of the temperature of some object is proportional to the difference between its own temperature and the temperature of its surroundings. For an object at temperature* T *placed in a closed system at temperature *T _{f}*, this change in temperature as a function of time

*t*is described by the differential equation:

*dT/dt = -k(T - T _{f })*,

**(Equation 2)**

*where k *is a constant that depends upon the characteristics of the object and its surroundings. **Equation 1 **is equivalently written as:

*-k dt = dT / (T - T _{f }). *

**(Equation 3)**

Integrating both sides gives:

-*k t = log(T - T _{f }) + log C. *

**(Equation 4)**

Applying the exponential function to both sides of the equation and then rearranging gives:

*T - T _{f }= C e^{-kt} . *

**(Equation 5)**

If the object in question is at an initial temperature *T _{i }*at time

*t =*0:

*T _{i}- T_{f } = C. *

**(Equation 6)**

It follows that the temperature as a function of time is:

*T(t) =T _{f }+ (T_{i}- T_{f }) e^{-kt }.*

**(Equation 7)**

Therefore, when a hot object is placed in a cooler closed system, its temperature will decrease at an exponential rate. In this closed system, the heat from the hot object *Q *will increase the temperature of the cooler surroundings and thus increase the number of available states. Thus, the change in entropy, Δ*S*,* *is positive and nonzero.

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### Procedure

1. Setup.

- Obtain a heating element and stand, a thermometer, a stopwatch, a few paper towels, water, and a large beaker.
- Fill the beaker with enough water so that the sample will not cool down too rapidly (
*i.e.,*at least 500 mL). - Place the beaker full of water on the stand below the heating element and turn it on.
- Once the beaker of water reaches a boil, insert the thermometer and turn off the heating element.
- Carefully remove the beaker from the heating stand and place it on the table, on top of the paper towels. These will act as insulation from the table.

2. Recording data.

- Begin the stopwatch and record the temperature and time.
- For the first 20 min, take a measurement about every 1 min.
- For the next 20 min, take a measurement about every 3-5 min.
- Record these values in
**Table 1**. - Plot the data points that were collected in
**Table 1**in a graph of temperature versus time*.* - Using the initial temperature of the water and any two data points for the time and temperature, solve
**Equation 7**for the cooling constant*k.* - Using this value for
*k,*plot**Equation 7**as a continuous function of*t*. Compare the function with the data points that were collected.

Entropy is a fundamental thermodynamic principle used to describe heat transfer in a system.

The term Entropy is often considered a measure of the "disorder" of a system and the second law of thermodynamics states that if the system is undergoing an irreversible process, then the entropy of the system will always increase.

Think about gas trapped in a container with known volume, pressure and temperature**.** The gas molecules can have an enormous number of possible configurations**.** If the container is opened, the gas molecules escape and the number of configurations increases dramatically, essentially approaching infinity. Therefore *S*, which denotes entropy, definitely increased after opening the container. Thus, Δ*S, *or the change in entropy,is greater than zero.

Similarly, entropy also increases when hot water is left at room temperature and allowed to cool down. In this video, we will illustrate how to measure the change in entropy of a system during such cooling experiments.

Before learning how to do the experiment and gather data, let's learn some laws and equations that allow us to calculate rate of temperature change and increase in entropy during cooling experiments.

Newton's Law of Cooling states that the rate of temperature change of an object is proportional to the difference between its own temperature and the temperature of the surroundings. Using calculus, this relationship can be converted into this equation, where lower case t represents time, Ts denotes temperature of the surroundings, T0 is the initial temperature, and k is a constant that depends on the characteristics of the object and its surroundings.

Using this equation, one can calculate the temperature of a cooling system at any time if all the other variables are known. This equation also shows that temperature is an exponential function of time. Thus, when a hot object, like a glass of hot water, is placed in a cooler environment, its temperature will decrease at an exponential rate until it reaches the temperature of the surroundings.

Now, let's see how to calculate the change in entropy, or Δ*S . *Let's rewind to when the water was hot.

When talking about entropy, we must first define the system. Here, the system is the glass of water plus the air in the room. So the change in entropy of the system, or Δ*Stotal *is a sum of the change in entropies of these individual components. Mathematically, the change in entropy is defined as heat gained or lost, denoted by Q, divided by the temperature.

In this scenario, we know that heat leaves water, thus Δ*S* for water decreases. On the contrary, the surrounding air gains heat. Therefore, Δ*S*air increases. From the second law of the thermodynamics, we know the change in entropy of the total system must be positive.

Now let's see how to conduct an experiment to test these theoretical predictions of Newton's Law of Cooling and the second law of thermodynamics.

To begin, fill a large beaker with between 500 mL to one L of water. Place the beaker on a hot plate, and heat the water to boiling. Once the water boils, turn off the heating element.

Then, carefully remove the beaker from the hot plate, and place it on the table on top of paper towels. The paper towels will act as insulation between the water and the cool table. Measure the temperature of the water using the thermometer.

Start the stopwatch, and record the temperature of the water every minute for the first 20 minutes.

For the next 20 minutes, record the temperature every 5 minutes.

Stop taking measurements when the water has come close to room temperature. Then, plot the data points in a graph of temperature of the water versus time.

Now let's analyze the data obtained. The initial temperature of the water was 100 degrees, at 35 minutes the temperature dropped to 50.6, and the surrounding temperature was 28.5 degrees. Plug in these values into Newton's Law of Cooling, and solve for the cooling constant *k*.

Now using the calculated value for *k*, plot the equation as a continuous function. If we lay our measured data points on this chart, we can see that the theoretical and experimental functions follow an almost identical path.

Now let's talk about entropy. As we know, the total change in entropy, or delta S, is equal to the entropy change for the water plus the room.

The change in entropy equals Q, or the amount of heat transferred from the hot water to the air, divided by T, so the change in entropy can be calculated if Q is known.

Q can be calculated using the relationship between mass, m, specific heat, c, and the change in temperature in Kelvin, delta T. Using the values for water the amount of heat released by the water, Q can be calculated and used to solve for delta S.

Thus, the experimental data shows that the entropy of the total system has increased since heat was transferred from the water to the air molecules in the room. This validates the second law of thermodynamics.

Entropy and the Second Law of Thermodynamics describe a wide range of occurrences in nature and engineering.

A refrigerator is essentially a heat pump, and removes heat from one location at a lower temperature, the heat source, and transfers it to another location, the heat sink, at a higher temperature.

According to the second law, heat cannot spontaneously flow from a colder location to a hotter one. Thus, work, or energy, is required for refrigeration.

A campfire is another example of entropy changes in real life. The solid wood used as fuel burns and turns into a disordered pile of ash. In addition, water molecules and carbon dioxide gas are released.

The atoms in the vapors spread out in an expanding cloud, with infinite disordered arrangements. Thus, the entropy change from burning wood is always positive.

You've just watched JoVE's introduction to entropy and the second law of thermodynamics. You should now understand the basic concept of entropy, Newton's Law of Cooling, and examples of entropy changes in everyday life. Thanks for watching!

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### Results

Representative results for 680 mL of water are shown in **Table 1**. The cooling constant *k *was found using the data points in the table and solving **Equation 7**. After 35 min, *T*(35) = 50.6. The initial temperature was 100 °C, and data collection ceased at 28.5 °C. Using these variables gives the following equation to obtain *k:*

50.6 = 28.5 + (100 - 28.5) *e*^{-k 35}. **(Equation 8)**

Solving for *k *gives a value *k *= 0.034. The curve with this cooling constant is shown as a dashed gray line in **Figure 1**, along with the data points from the experiment. The functional form of **Equation 6 **matches the experimental results very closely.

As the water cools, the entropy decreases, since the number of states available to the water molecules decreases. The entropy of the ambient air in the room increases because the water beaker transfers heat to the air molecules surrounding it; the overall entropy of the water + air system increases. The number of states that the now-hotter air molecules can occupy is much higher than before the hot water was introduced to the room.

In differential form, the heat d*Q *added to or removed from the water can be calculated using the relationship between mass, specific heat *c*, and temperature change:

d*Q = mc *d*T*, **(Equation 9)**

where *c* is known to be 4.18 J/(gK) for water. The change in entropy of the water is then:

Δ*S _{water} = *

= *m*_{water }*c*_{water }ln(T_{final} / T_{initial}). **(Equation 10)**

Using the conversion to Kelvin as K = °C^{ }+ 273.15, the change in the entropy of the water is calculated as:

Δ*S _{water} = *680 g * 4.18 J/(g K)

***ln[(28.5 + 273.15)/(100 + 273.15)]

= -604 J/K.

The ambient air temperature is constant at 20.4 °C, so this is an isothermal process. The entropy change of the air is then:

Δ*S _{air} = *,

where *Q *is the heat released by the water, which is given by **Equation 9**. The change in entropy of the air is then calculated as:

Δ*S _{air} = *

= 3337 J/K.

The total change in the entropy of the water + air system, Δ*S _{tot}*, is the sum of the individual changes in entropy of the water and ambient air:

Δ*S _{tot} = ΔS_{water} +ΔS_{air}^{ }*

**(Equation 11)**

= -604 J/k + 3337 J/K

= 2733 J/K.

**Table 1. Temperatures recorded during the experiment.**

Time (min s) | Water Temperature (°C) |

0 0 | 99.6 |

1 10 | 97.1 |

1 50 | 94.2 |

2 30 | 91.8 |

3 22 | 89 |

4 05 | 87.2 |

5 08 | 82.7 |

6 05 | 82.4 |

8 25 | 78 |

9 15 | 76.5 |

10 15 | 74.6 |

11 38 | 72.7 |

12 58 | 70.7 |

13 58 | 69.2 |

15 15 | 67.7 |

16 55 | 65.8 |

18 38 | 64 |

20 25 | 62.3 |

24 02 | 58.8 |

25 45 | 57.3 |

34 45 | 50.6 |

40 50 | 47.4 |

44 30 | 45.9 |

49 59 | 43.6 |

53 42 | 42.4 |

60 01 | 40.2 |

64 20 | 39.5 |

76 37 | 37 |

103 50 | 32.1 |

116 41 | 30.3 |

122 46 | 29.6 |

134 11 | 28.5 |

**Figure 1****. Plot of temperature versus time.** The blue dots indicate the experimental data, and the dashed line represents theoretical data based on Newton's law of cooling.

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### Applications and Summary

A pair of headphones kept in a bag always tends to become knotted-this is an increase in entropy caused by carrying the bag around. It is necessary to do work on the headphones to un-knot them and decrease the entropy (this can be thought of as a "reversible process"). The most efficient heat engine cycle allowed by physical laws is the Carnot cycle. The second law states that not all heat supplied to a heat engine can be used to do work. The Carnot efficiency sets the limiting value on the fraction of heat that can be used. The cycle consists of two isothermal processes followed by two adiabatic processes. A refrigerator, which is essentially just a heat pump, is also a classic example of the second law. Refrigerators move heat from one location at a lower temperature (the "source") to another location at a higher temperature (the "heat sink") using mechanical work. According to the second law, heat cannot spontaneously flow from a colder location to a hotter one; thus, work (energy) is required for refrigeration.

Newton's law of cooling was demonstrated by a beaker full of water at 100 °C cooling down to room temperature, which led to an increase in the entropy of the water-air system. By measuring the temperature of the water as a function of time over a period of 135 min, it was possible to confirm that the cooling of the water was exponential in form. The cooling constant of the water sample was found by solving the cooling equation using collected data.

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