Question

If two objects, A and B, of different temperature come into direct contact, what is the relationship between the heat lost by one object and the heat gained by the other? What is the relationship between the temperature changes of the two objects? (Assume that the two objects do not lose any heat to anything else.)

Short answer

The heat lost by object A is equal to the heat gained by object B, assuming no heat is lost to the surroundings. The temperature changes of the objects depend on their specific heat capacities and masses; the object with a lower product of mass and specific heat capacity will undergo a larger temperature change.

Step-by-step solution

Identifying the Law of Conservation of Energy

Understand that the Law of Conservation of Energy states that energy cannot be created or destroyed in an isolated system. When two objects of different temperatures come into contact, energy in the form of heat will flow from the warmer object to the cooler one until thermal equilibrium is reached.

Relating Heat Loss and Heat Gain

Apply the principle that the amount of heat lost by the warmer object (A) must be equal to the amount of heat gained by the cooler object (B). This can be represented by the equation: \(Q_{loss} = Q_{gain}\), where \(Q_{loss}\) is the heat lost by object A, and \(Q_{gain}\) is the heat gained by object B.

Understanding the Relationship Between Temperature Changes

Relate the temperature change of the objects to their heat transfer using the formula: \(Q = mc\Delta T\), where Q is the heat transferred, m is the mass of the object, c is the specific heat capacity, and \(\Delta T\) is the change in temperature. Since the heat lost by one object is equal to the heat gained by the other, the amount of heat (\(Q\)) is the same for both. If the masses and specific heat capacities of the two objects are the same, they would experience equal and opposite changes in temperature. Otherwise, the changes in temperatures will be inversely proportional to the products of their masses and specific heat capacities.

Key concepts

Heat Transfer

Heat transfer is a fundamental concept that describes the movement of thermal energy from one place or material to another, due to a temperature difference. It occurs in three main ways: conduction, when heat moves through a solid; convection, in which fluid movement facilitates heat spread; and radiation, where heat travels as electromagnetic waves.

Imagine two metal rods, one hot and the other cold, touching at one end. The heat naturally flows from the hot rod to the colder one, aiming to equalize the temperatures—a process called thermal conduction. Factors affecting the rate of this heat transfer include temperature difference, properties of the materials involved like thermal conductivity, and the surface area over which the contact occurs.

Understanding this concept helps to explain why, in the exercise, the warmer object loses heat to the cooler object. It's a natural balancing act aimed at reaching a state of equilibrium. In a practical sense, insulating materials exploit this principle by slowing down heat transfer to keep beverages hot or cold, or homes warm.

Thermal Equilibrium

Thermal equilibrium is a state reached when two or more objects in contact with each other cease to exchange heat, leading to a uniform temperature among them. A simple example could be a mug of coffee left on a table. Initially, the coffee, being hot, will transfer heat to the surrounding cooler air. Over time, the coffee cools down, and the system reaches a point where the temperature of the coffee and the surrounding air equalizes, resulting in no further heat transfer.

This concept is grounded in the Zeroth Law of Thermodynamics, which states that if two systems are each in thermal equilibrium with a third system, they are also in equilibrium with each other. Applying this to the textbook exercise, once the two objects have exchanged heat energy, they will eventually settle at a shared temperature, provided they remain isolated from any other influences.

Thermal equilibrium is pivotal in processes such as climate control in buildings and maintaining the operational temperature of machinery and electronic devices to ensure optimal performance and prevent overheating.

Specific Heat Capacity

The specific heat capacity is a property that measures the amount of heat energy required to raise the temperature of a unit mass of a substance by one degree Celsius (or one Kelvin). It is a substance-specific value that plays a critical role in how quickly or slowly a material heats up or cools down when exchanging heat.

For example, water has a high specific heat capacity, which means it can absorb a lot of heat before it starts to get warmer. This characteristic of water helps regulate the Earth's climate and also makes it an excellent coolant in various technologies.

In the context of the textbook exercise, two objects with different specific heat capacities will experience unequal temperature changes, even if the amount of heat exchanged is the same. An object with a lower specific heat capacity will undergo a more significant temperature change than one with a higher capacity. This concept elucidates why different materials heat up or cool down at different rates, a consideration vital in designing thermal systems, like radiators or heat sinks, and in culinary arts, such as understanding how to properly temper chocolate.