Problem 5 - Astronomy Journey to the Cosmic Frontier

Question

Suppose the temperature of a body is higher than the temperature at which it would be in thermal equilibrium. Describe how and why the temperature of the body will change with time.

Step-by-Step Solution

Answer

The temperature of the body decreases over time until it reaches ambient temperature due to heat transfer.

Step 1: Understanding Thermal Equilibrium

Thermal equilibrium is the state when two or more bodies exchange energy related to heat until their temperatures equalize. An object not in equilibrium will lose or gain heat over time.

Step 2: Applying Newton's Law of Cooling

When a body is hotter than its surroundings, it tends to lose heat according to Newton's Law of Cooling, which states that the rate of heat loss is proportional to the temperature difference between the body and its environment.

Step 3: Temperature Change Over Time

As the body cools down, its temperature decreases following Newton's Law of Cooling: \[\frac{dT}{dt} = -k(T - T_{ambient})\]where \(T\) is the object's temperature, \(T_{ambient}\) is the ambient temperature, and \(k\) is a constant depending on the characteristics of the object.

Step 4: Reaching Thermal Equilibrium

The body will continue to lose heat until its temperature is equal to the ambient temperature. This is when thermal equilibrium is reached, and the temperature change will stop because there is no longer a temperature gradient.

Key Concepts

Newton's Law of Cooling

Newton's Law of Cooling is a pivotal concept in understanding how temperatures of objects change over time when exposed to different environments. This law states that the rate at which an object loses (or gains) heat is directly proportional to the temperature difference between the object and its surrounding environment. This is mathematically expressed as:\[ \frac{dT}{dt} = -k(T - T_{ambient}) \]In this equation:

• \(dT/dt\) is the rate of change of temperature over time.
• \(T\) is the temperature of the object.
• \(T_{ambient}\) is the ambient or surrounding temperature.
• \(k\) is a proportionality constant specific to the object.

The negative sign indicates that the temperature of the object decreases over time. The larger the difference between the object's temperature and the ambient temperature, the faster the rate of cooling. This principle is widely applicable, from cooling cups of coffee to the temperature changes in more complex systems like atmospheric conditions.

Temperature Change

Temperature change is a dynamic process wherein a hot object cools down over time until it equilibrates with the temperature of its surroundings. The temperature change is governed by Newton's Law of Cooling, which implies that the rate of cooling is not constant but instead decreases as the temperature difference between the object and its environment decreases. The key takeaway from the formula \( \frac{dT}{dt} = -k(T - T_{ambient}) \) is that the rate of temperature change depends heavily on the initial temperature difference:

• Larger initial temperature differences result in faster cooling rates.
• As the temperature approaches ambient temperature, the rate of cooling slows down progressively.

Over time, the effect of this process results in a gradual decrease in the temperature of the object until it becomes equal to the ambient temperature.

Ambient Temperature

Ambient temperature plays a critical role in determining how quickly an object reaches thermal equilibrium. It is essentially the temperature of the surrounding environment and acts as a benchmark towards which the temperature of any object tends. In the context of Newton's Law of Cooling, ambient temperature is the temperature at which the cooling process aims to stabilize. Key aspects of ambient temperature include:

• It serves as a stable reference point for temperature change.
• The greater the difference between ambient temperature and an object's temperature, the more rapid the cooling or heating process.
• Once an object's temperature aligns with the ambient temperature, it is said to have reached thermal equilibrium, putting a halt to further temperature change.

Understanding ambient temperature is essential because it helps predict how and when objects will stop undergoing temperature changes, solidifying the concept of reaching thermal equilibrium.