Question

A body at an initial temperature of \(T_{i}\) is brought into a medium at a constant temperature of \(T_{\infty}\). How can you determine the maximum possible amount of heat transfer between the body and the surrounding medium?

Short answer

Answer: The maximum possible heat transfer (Qmax) between a body and a surrounding medium can be calculated using the following equation: Qmax = mc(T∞ - Ti).

Step-by-step solution

Understanding the Heat Transfer Process

The process of heat transfer between the body and the medium can be represented by the First Law of Thermodynamics. It states that the amount of heat transferred to a system is the difference between the internal energy of the system at the final state and the internal energy at the initial state. In this case, we can write the First Law of Thermodynamics as: \(Q = mc(T_f - T_i)\) where \(Q\) is the amount of heat transfer, \(m\) is the mass of the body, \(c\) is the constant heat capacity of the body, \(T_f\) is the final temperature of the body, and \(T_i\) is the initial temperature of the body.

Determine the Final Temperature (Maximum Heat Transfer)

To achieve the maximum heat transfer, the final temperature of the body must be equal to the constant temperature of the medium, that is, \(T_f = T_\infty\). Thus, we can rewrite the equation as: \(Q_{max} = mc(T_\infty - T_i)\)

Calculate the Maximum Possible Heat Transfer

Now that we have the equation relating the maximum heat transfer to the temperature difference and the heat capacity, we can calculate the maximum possible amount of heat transfer between the body and the surrounding medium. By substituting the values of \(T_i\), \(T_\infty\), \(m\), and \(c\) (if given or known) into the equation, we can find the value of \(Q_{max}\), which represents the maximum possible amount of heat transfer between the body and the medium.

Key concepts

First Law of Thermodynamics

The First Law of Thermodynamics is a fundamental principle that serves as a cornerstone of thermal physics. It asserts that energy cannot be created or destroyed within an isolated system; rather, it can only be transformed from one form to another. In the context of heat transfer, this law posits that the change in the internal energy of a system is equivalent to the heat added to the system minus the work done by the system on its surroundings.

In essence, when we consider a body being heated or cooled, the First Law tells us that the heat we are measuring has transferred into or out of the body, thereby changing its internal energy. The formula that arises from this principle is given by:
\[\Delta U = Q - W\]
Here, \(\Delta U\) represents the change in internal energy, \(Q\) denotes the heat exchanged, and \(W\) is the work done by or on the system. For the scenario of our exercise, since the body is not performing work, the equation simplifies to:
\[Q = \Delta U\]
This expression allows us to calculate the heat transferred based solely on changes in internal energy. Remember, if we know the final state the body must reach for maximum heat transfer, we can then use this law to determine the corresponding change in internal energy.

Heat Capacity

Heat capacity is a critical concept in thermodynamics that measures the amount of heat required to increase the temperature of a given mass by one degree Celsius (or one Kelvin). It dictates how much heat energy is needed for a particular object to undergo a temperature change. The heat capacity of a substance is determined by both its mass (often denoted as \(m\)) and its specific heat capacity (commonly represented as \(c\)), an intrinsic property that indicates the substance's ability to absorb heat per unit mass per degree temperature change.

Mathematically, the heat capacity \(C\) of a body is described by the formula:
\[C = mc\]
This formula emphasizes the direct proportionality between heat capacity and both the mass and specific heat of a substance. The larger the heat capacity, the more heat is required to alter the temperature of a body—this can be crucial when dealing with thermal processes. For the exercise in question, understanding heat capacity aids in calculating the precise amount of heat transferred during the temperature shift from the initial state to the equilibrium state with the surrounding medium.

Temperature Difference

Temperature difference is an elemental concept when examining heat transfer processes. It's the driving force that propels heat flow from one area to another, particularly from a hotter object to a cooler one, until thermal equilibrium is reached. In any heat transfer calculation, the difference in temperature between two points—in our exercise, the initial temperature of the body \(T_i\) and the constant temperature of the medium \(T_\infty\)—is essential.

The greater the temperature difference, the greater the potential for heat transfer. This notion underpins the simple yet critical equation for heat transfer:
\[Q = mc(T_f - T_i)\]
which involves the specific heat capacity \(c\) of the material, the mass \(m\), and the change in temperature \((T_f - T_i)\). In scenarios where maximum heat transfer is desired, as in our textbook exercise, the final temperature \(T_f\) of the body is considered to be equal to the medium's constant temperature \(T_\infty\), thus maximizing the temperature difference if the initial temperature is lower.

Internal Energy

Internal energy is a thermodynamic term that encompasses the total energy contained within a system, which is due to the kinetic and potential energies of the molecules within the system. It is an intrinsic property of the system that changes with alterations in temperature, volume, or pressure.

When a system undergoes a heat transfer without performing any external work, as is posited in our exercise, the heat transfer directly correlates to a change in the internal energy. The relationship between heat transfer and internal energy, without work being involved, can be expressed as:\(Q = \Delta U\), which is a specific form of the First Law of Thermodynamics for isochoric processes (where volume remains constant).

In practical terms, by understanding the amount of heat required or released to change the internal energy, we can estimate how a system's temperature will alter. Thus, for the maximum heat transfer to occur, the internal energy of the body must increase (when absorbing heat) or decrease (when releasing heat) to align with the internal energy corresponding to the temperature of the surrounding medium.