Problem 44 - Physics

Question

A piece of glass has a temperature of \(83.0^{\circ} \mathrm{C}\). Liquid that has a temperature of \(43.0^{\circ} \mathrm{C}\) is poured over the glass, completely covering it, and the temperature at equilibrium is \(53.0^{\circ}\) C. The mass of the glass and the liquid is the same. Ignoring the container that holds the glass and liquid and assuming that the heat lost to or gained from the surroundings is negligible, determine the specific heat capacity of the liquid.

Step-by-Step Solution

Answer

The specific heat capacity of the liquid is 2.52 J/g°C.

Step 1: Define the Known Variables

We need to identify the important given variables for this problem. \(T_\text{initial, glass} = 83.0^\circ C\), \(T_\text{initial, liquid} = 43.0^\circ C\), \(T_\text{equilibrium} = 53.0^\circ C\), and the mass of the glass \(m_g = m_l\) (same as the mass of the liquid).

Step 2: Express the Heat Transfer Equations

The heat lost by the glass \(Q_g = m_g c_g (T_\text{initial, glass} - T_\text{equilibrium})\) and the heat gained by the liquid \(Q_l = m_l c_l (T_\text{equilibrium} - T_\text{initial, liquid})\). Since the mass is the same, use \(m\) for both.

Step 3: Use the Conservation of Energy Principle

Since no heat is lost to the surroundings, the heat lost by the glass is equal to the heat gained by the liquid: \[Q_g = Q_l\]\[m c_g (T_\text{initial, glass} - T_\text{equilibrium}) = m c_l (T_\text{equilibrium} - T_\text{initial, liquid})\].

Step 4: Substitute Known Values and Simplify

Substitute the known values: \[c_g (83.0 - 53.0) = c_l (53.0 - 43.0)\]\[c_g \times 30.0 = c_l \times 10.0\].

Step 5: Insert the Specific Heat Capacity of Glass and Solve

The specific heat capacity of glass is \(c_g = 0.84\, \text{J/g°C}\). Replace \(c_g\) in the equation: \[0.84 \times 30.0 = c_l \times 10.0\]\[25.2 = 10 c_l\].

Step 6: Solve for the Specific Heat Capacity of the Liquid

Divide both sides by 10 to find \(c_l\): \[c_l = \frac{25.2}{10} = 2.52\, \text{J/g°C}\].

Key Concepts

Heat Transfer

Heat transfer is the process of thermal energy moving from a hotter object to a cooler one. This phenomenon occurs due to the temperature difference between the two entities. In our example, the glass is warmer at the start, while the liquid is cooler.

• As soon as these two items come into contact, heat begins to transfer from the glass to the liquid.
• The ultimate goal is to reach a balance, where both objects attain the same temperature, known as thermal equilibrium.

The heat transfer continues until there's no longer a temperature difference, thus achieving equilibrium.

Energy Conservation

The principle of energy conservation states that energy in an isolated system cannot be created or destroyed. In the context of calorimetry, this means that heat lost by one material must be equal to the heat gained by another.

• In the given problem, heat lost by the glass is equal to the heat gained by the liquid since no heat is lost to the surroundings.
• This ensures that total energy is conserved throughout the process.

Mathematically, it means:\[m c_g (T_{\text{initial, glass}} - T_{\text{equilibrium}}) = m c_l (T_{\text{equilibrium}} - T_{\text{initial, liquid}})\]

Thermal Equilibrium

Thermal equilibrium is reached when two substances within a closed system reach the same temperature. At this point, no heat flows between them because the temperature is uniform. In the example provided, the glass and liquid reach a common final temperature of \(53.0^{\circ}C\).

• This new temperature indicates that the system has reached thermal equilibrium.
• The lack of additional heat flow suggests that energy distribution between the glass and the liquid is balanced.

Understanding thermal equilibrium allows scientists to make calculations about the specific heat capacities of various materials.

Calorimetry

Calorimetry is the science of measuring the amount of heat involved in a chemical or physical process. By observing changes in temperature, we can infer energy exchanges and understand the properties of substances.

• In the exercise, calorimetry techniques help us determine the specific heat capacity of the liquid in question.
• By applying the concept of energy conservation and knowing the specific heat capacity of the glass, we can solve for the unknown specific heat capacity of the liquid.

This is done using the formula:\[c_l = \frac{c_g \times (T_{\text{initial, glass}} - T_{\text{equilibrium}})}{(T_{\text{equilibrium}} - T_{\text{initial, liquid}})}\]Calorimetry helps in understanding heat transfer in chemical reactions and changes in matter.