Question

An office worker claims that a cup of cold coffee on his table warmed up to \(80^{\circ} \mathrm{C}\) by picking up energy from the surrounding air, which is at \(25^{\circ} \mathrm{C}\). Is there any truth to his claim? Does this process violate any thermodynamic laws?

Short answer

Why or why not? Answer: No, it is not possible for the cold coffee to warm up to 80°C from the surrounding air at 25°C without any external energy input. This is because it would violate the second law of thermodynamics, which states that the total entropy of an isolated system can only increase over time. In this case, the heat transfer process would result in a decrease in the total entropy, contrary to the second law of thermodynamics.

Step-by-step solution

Determine if heat transfer is occurring

In this situation, the cup of coffee would warm up and the surrounding air would lose energy and cool down. We can conclude that there must be heat transfer between the coffee and the surrounding air.

Determine the direction of heat transfer

Heat transfer occurs from a higher temperature body to a lower temperature body. This means that for heat transfer to occur, the initial temperature of the coffee must be lower than the temperature of surrounding air. However, their initial temperatures are not given. Hence, we must look into the process in a more general way.

Analyze the second law of thermodynamics

The second law of thermodynamics states that the total entropy of an isolated system can only increase over time. Mathematically, for a heat transfer process, it can be written as: \(dS \geq \frac{Q}{T}\) Where \(dS\) is the change in entropy, \(Q\) is the heat transfer, and \(T\) is the temperature of the system. If \({Q}_{coffee}>0\), then the coffee is absorbing heat, and \({Q}_{air}=-{Q}_{coffee}<0\). If \({Q}_{coffee}<0\), it means that the coffee is losing heat and \({Q}_{air}=-{Q}_{coffee}>0\). In both cases, we must look at the entropy changes for coffee and air, and the sum of their changes should be non-negative for the process to obey the second law of thermodynamics.

Check the entropy changes

The entropy change for the coffee can be written as \({\Delta S}_{coffee} \geq \frac{{Q}_{coffee}}{T_{coffee}}\), and the entropy change for the air can be written as \({\Delta S}_{air} \geq \frac{{Q}_{air}}{T_{air}}\). Now we analyze the situation in which the coffee warms up to \(80^{\circ} \mathrm{C}\), and the surrounding air is at \(25^{\circ} \mathrm{C}\): If the coffee absorbs heat (\(Q_{coffee}>0\)) from the surrounding air, we have: \({\Delta S}_{coffee} \geq \frac{{Q}_{coffee}}{80+273}\) \({\Delta S}_{air} \geq -\frac{{Q}_{coffee}}{25+273}\) Adding these two inequalities, we get: \({\Delta S}_{total} = {\Delta S}_{coffee} + {\Delta S}_{air} \geq {Q}_{coffee}\left(\frac{1}{353}-\frac{1}{298}\right)\) Notice that the term inside the parenthesis is negative. Since \({Q}_{coffee}>0\), \({\Delta S}_{total}<0\), which contradicts the second law of thermodynamics. This means that it is impossible for the coffee to warm up to \(80^{\circ} \mathrm{C}\) from the surrounding air at \(25^{\circ} \mathrm{C}\) without any external energy input. In conclusion, the office worker's claim that the cold coffee warmed up to \(80^{\circ} \mathrm{C}\) by picking up energy from the surrounding air at \(25^{\circ} \mathrm{C}\) is false, as it violates the second law of thermodynamics.

Key concepts

Understanding Heat Transfer

Heat transfer is the movement of thermal energy from one object or substance to another. It's a fundamental concept in thermodynamics and plays a crucial role in the example of the office worker's coffee cup. In an everyday context, we observe heat transfer when a warm object cools down or a cool object warms up, as thermal energy moves from the warmer object to the cooler one.

In the scenario with the coffee warming up from the surrounding air, heat transfer would need to occur for the coffee's temperature to change. However, this transfer naturally occurs from a hot body to a cooler one—never the reverse, unless work is done on the system. This premise is rooted in the directionality of heat flow, which is inherently linked to the temperatures of the objects involved. If the coffee were cooler than the surrounding air, we would expect heat to transfer from the air to the coffee. Contrariwise, if the coffee were already hotter than the environment, no spontaneous heat transfer would occur to further raise its temperature.

Understanding the direction of heat transfer allows us to evaluate if certain events, like the claim about the coffee warming to a temperature higher than its surroundings, are physically plausible.

The Role of Entropy Change in Thermodynamics

Entropy change is a measure of disorder or randomness within a thermodynamic system and reflects the number of ways a system can be arranged, often leading to the dispersion of energy. It is a central concept in the second law of thermodynamics, which broadly suggests that entropy in an isolated system tends to increase over time. This means that the total change in entropy, when considering all parts of a system, must be equal to or greater than zero in any spontaneous process.

In the context of the coffee and the surrounding air, if heat were transferred from the air to the coffee, the coffee's entropy would increase—since it's gaining energy—and the air's entropy would decrease, as it loses energy. However, the second law tells us that the total entropy change, the sum of the entropy changes in the coffee and the air, must not be negative. This is why the claim about the coffee spontaneously warming up to a higher temperature than its surroundings is flawed. A positive heat transfer to the coffee would lead to a negative total entropy change, contradicting the second law of thermodynamics.

When tackling thermodynamic problems, carefully considering the entropy change involved in heat transfers is essential to ensure that the proposed scenario aligns with the fundamental principles of physics.

Thermodynamic Laws and Their Implications

Thermodynamic laws provide a framework for understanding how energy is conserved and transferred in physical systems. They outline the limitations and possibilities within thermodynamics, guiding us in predicting the behavior of systems under various conditions.

The first law of thermodynamics, also known as the law of energy conservation, states that energy cannot be created or destroyed, only transformed from one form to another. This law governs the principle that the total energy of an isolated system remains constant. The second law, as we've discussed with reference to the coffee scenario, introduces the concept of entropy and gives insight into the directionality of spontaneous processes.

These laws collectively reveal that the coffee cannot spontaneously warm up to a higher temperature than that of the surrounding air without additional work or energy input. To claim otherwise would be to ignore these fundamental principles that have been validated by extensive empirical evidence. In thermodynamics exercises, it's crucial for students to understand not just the individual laws but also their synergy, as they form the bedrock upon which all thermodynamic phenomena are understood.