Question

Suppose object \(A\) has twice the specific heat and twice the mass of object \(B\). If the same amount of heat is applied to both objects, how will the temperature change of \(A\) be related to the temperature change in \(B\) ?

Short answer

The temperature change of object A, \( \Delta T_A \), will be one fourth that of object B, \( \Delta T_B \).

Step-by-step solution

Identify Given Information

The specific heat capacity of object A is twice that of object B, which we will denote as \( C_A = 2C_B \). The mass of object A is also twice that of object B, denoted \( m_A = 2m_B \). The same quantity of heat \( Q \) is added to both objects.

Apply the Heat Transfer Equation

We use the equation for heat transfer \( Q = mc\Delta T \), where \( m \) is mass, \( c \) is the specific heat capacity, and \( \Delta T \) is the temperature change. For both objects receiving the same amount of heat, we set up two equations: \[ Q = m_A C_A \Delta T_A \] and \[ Q = m_B C_B \Delta T_B \].

Substitute the Known Ratios

Substitute the known relationships into the equations: \( m_A = 2m_B \) and \( C_A = 2C_B \) to get \( Q = (2m_B)(2C_B)\Delta T_A \) and \( Q = m_B C_B \Delta T_B \).

Solve for the Ratio of Temperature Changes

Divide both sides of the first equation by \( 2m_B C_B \) to isolate \( \Delta T_A \): \[ \frac{Q}{2m_B C_B} = 2\Delta T_A \]. Doing the same for the second equation gives \[ \frac{Q}{m_B C_B} = \Delta T_B \]. Now, we can divide the equation for \( \Delta T_A \) by the equation for \( \Delta T_B \) to find the ratio: \[ \frac{2\Delta T_A}{\Delta T_B} = \frac{1}{2} \], meaning \( \Delta T_A = \frac{1}{4}\Delta T_B \).

Key concepts

Heat Transfer

Heat transfer is a fundamental concept in physics and deals with the movement of thermal energy between different bodies and within material substances. It can occur in three ways: conduction, in which heat is transferred via direct contact between materials; convection, where heat is moved by the fluid movement within a gas or liquid; and radiation, where heat is transferred in the form of electromagnetic waves.

In the context of our exercise, we’re focusing on how heat transfer impacts the temperature change in solid objects, specifically through heat conduction. When a certain amount of heat is applied to an object, the temperature of the object increases. This process can be quantified using the heat transfer equation \( Q = mc\Delta T \), where \( Q \) represents the heat added, \( m \) is the mass of the object, \( c \) is the specific heat capacity, and \( \Delta T \) is the change in temperature.

The relationship between these variables is essential for understanding how different materials respond to heat. The specific heat capacity \( c \) is particularly important because it indicates how much heat is required to raise the temperature of one gram of the substance by one degree Celsius. Materials with higher specific heat capacities will require more heat to achieve the same temperature change as materials with lower specific heat capacities, a central concept in the given exercise.

Temperature Change

Temperature change represents the variation in the thermal state of an object, often resulting from heat transfer processes. The heat transfer equation, \( Q = mc\Delta T \), describes the direct relationship between the heat added to a substance and its resulting temperature change.

Considering our textbook exercise, we understand that the specific heat and mass of object \(A\) are both twice that of object \(B\). This means that for the same amount of heat \(Q\), object \(A\) will experience a smaller temperature change compared to object \(B\) due to its larger mass and higher specific heat. Mathematically, this translates to \(\Delta T_A = \frac{1}{4}\Delta T_B\), which reveals that object \(A\)'s temperature change is a quarter of object \(B\)'s.

To ensure comprehensive understanding, it's crucial to remember that the specific heat capacity acts as a 'buffer' against temperature change. This concept explains why materials like water, with high specific heat, are excellent at moderating temperature—they absorb or release more heat without experiencing drastic temperature changes. Therefore, in applications that involve heating or cooling, knowing the specific heat capacity is vital for controlling temperature effectively.

Thermodynamics

Thermodynamics is the branch of physics that deals with heat and temperature and their relation to energy and work. It encompasses all aspects of heat transfer and temperature change, as well as the laws that govern these processes. 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.

In the scope of our exercise, when heat \(Q\) is added to the objects \(A\) and \(B\), the thermal energy increases, and consequently, the objects' temperatures rise. The process honors the first law of thermodynamics, as the energy provided to the system is conserved and converted to an increase in thermal energy, leading to a temperature increase.

Furthermore, thermodynamics tells us why object \(A\) experiences a smaller temperature change despite receiving the same amount of heat as object \(B\). Object \(A\) has a higher storage capacity for thermal energy due to its greater mass and specific heat capacity. This example illustrates the application of thermodynamic principles in predicting how objects behave when subjected to heat. Understanding the fundamental concepts of thermodynamics is essential for students as it lays the groundwork for more advanced topics in physics and engineering, such as energy production, refrigeration, and heat engine designs.