Question

Two iron bolts of equal mass-one at \(100 .{ }^{\circ} \mathrm{C},\) the other at \(55^{\circ} \mathrm{C}\) - are placed in an insulated container. Assuming the heat capacity of the container is negligible, what is the final temperature inside the container \((c\) of iron \(=0.450 \mathrm{~J} / \mathrm{g} \cdot \mathrm{K}) ?\)

Short answer

The final temperature inside the container is 77.5°C.

Step-by-step solution

- Understand the concept

When two objects of different temperatures are placed in an insulated container, they will exchange heat until they reach thermal equilibrium. This means they will have the same final temperature.

- Set up the heat exchange equation

Since the two bolts have equal mass and they will share their heat equally, the amount of heat lost by the hotter bolt will be equal to the amount of heat gained by the cooler bolt. We can use the equation:\[ c \times m \times (T_{final} - T_{hot}) + c \times m \times (T_{final} - T_{cold}) = 0 \]where \( T_{hot} \) is the initial temperature of the hot bolt (100°C), \( T_{cold} \) is the initial temperature of the cold bolt (55°C), \( c \) is the specific heat capacity of iron, and \( m \) is the mass of each bolt.

- Simplify the equation

Since the masses and specific heat capacities are the same, they can be factored out of the equation. This results in:\[ (T_{final} - T_{hot}) + (T_{final} - T_{cold}) = 0 \]

- Solve for the final temperature

Rearrange the simplified equation to solve for \( T_{final} \):\[ 2T_{final} - (T_{hot} + T_{cold}) = 0 \]\[ 2T_{final} = T_{hot} + T_{cold} \]\[ T_{final} = \frac{T_{hot} + T_{cold}}{2} \]Substitute the initial temperatures into the equation:\[ T_{final} = \frac{100 + 55}{2} = \frac{155}{2} = 77.5 \text{°C} \]

Key concepts

Heat Exchange

When two objects with different temperatures come into contact inside an insulated container, they undergo a process known as heat exchange. The object with the higher temperature transfers heat to the one with the lower temperature. Over time, this transfer continues until both objects reach the same final temperature, achieving thermal equilibrium. This means the exchange of thermal energy between the two objects will cease as they become balanced.

Specific Heat Capacity

Specific heat capacity is a measure of how much heat energy is needed to change the temperature of a substance. It is denoted by the symbol 'c' and expressed in units of J/(g·K). In our example, the specific heat capacity of iron is given as 0.450 J/(g·K). This means that 0.450 joules of energy are required to raise or lower the temperature of 1 gram of iron by 1 degree Kelvin. Understanding this concept is crucial because it helps determine how much energy is exchanged during the process of reaching thermal equilibrium.

Thermodynamics

Thermodynamics is the branch of physics that deals with the principles of heat and energy transfer. In our exercise, the key concept of thermodynamics applicable is the principle of conservation of energy. According to this principle, the total energy in an isolated system remains constant. Thus, the heat lost by the hot iron bolt must equal the heat gained by the cold iron bolt. This forms the basis of setting up and solving the heat exchange equation in our solution.

Insulated Systems

An insulated system prevents the loss or gain of heat from its surroundings. In the context of our problem, the container holding the two iron bolts is insulated, meaning no heat escapes into the environment. Such a setup ensures that all the heat exchanged between the two bolts stays within the system, simplifying our calculations. It makes it possible to assume that the heat lost by the hotter bolt is completely gained by the cooler bolt.

Temperature Calculation

The final temperature can be calculated using the principle of heat exchange in combination with the specific heat capacity of the materials involved. Here’s how it works step-by-step: First, we set up the heat exchange equation, taking into account that the heat lost by the hotter bolt equals the heat gained by the cooler bolt. For simplicity, this equation can be written as: \[c \times m \times (T_{final} - T_{hot}) + c \times m \times (T_{final} - T_{cold}) = 0\] Given the masses and specific heat capacities are the same for both bolts, these terms cancel out: \[(T_{final} - T_{hot}) + (T_{final} - T_{cold}) = 0\] We then solve the simplified equation for the final temperature \(T_{final}\): \[2T_{final} - (T_{hot} + T_{cold}) = 0\] By substituting the initial temperatures (100°C and 55°C) of the two bolts, we find that: \[T_{final} = \frac{T_{hot} + T_{cold}}{2} = \frac{100 + 55}{2} = 77.5 °C\] Thus, the final temperature at thermal equilibrium is 77.5°C.