Problem 39 - University Physics with Modern Physics

Question

\(\mathrm{A} 500.0-\mathrm{g}\) chunk of an unknown metal, which has been in boiling water for several minutes, is quickly dropped into an insulating Styrofoam beaker containing 1.00 \(\mathrm{kg}\) of water at room temperature \(\left(20.0^{\circ} \mathrm{C}\right) .\) After waiting and gently stirring for 5.00 minutes, you observe that the water's temperature has reached a constant value of \(22.0^{\circ} \mathrm{C}\) (a) Assuming that the Styrofoam absorbs a negligibly small amount of heat and that no heat was lost to the surroundings, what is the specific heat of the metal? (b) Which is more useful for storing thermal energy: this metal or an equal weight of water? Explain. (c) What if the heat absorbed by the Styrofoam actually is not negligible. How would the specific heat you calculated in part (a) be in error? Would it be too large, too small, or still correct? Explain.

Step-by-Step Solution

Answer

Specific heat of metal: \(214.36 \ J/(kg \cdot °C)\). Water is more useful for thermal storage. Neglecting Styrofoam heat leads to underestimated metal specific heat.

Step 1: Understand the Heat Exchange Process

In this process, the heat lost by the metal equals the heat gained by the water and the Styrofoam. Given that the Styrofoam absorbs negligibly small heat, we will consider only the water. The formula is: \[ q_{metal} = -q_{water} \]

Step 2: Set Up the Heat Equation

The heat gained by the water and the heat lost by the metal are equal in magnitude:\[ m_{metal} \cdot c_{metal} \cdot (T_{final} - T_{initial, metal}) = -m_{water} \cdot c_{water} \cdot (T_{final} - T_{initial, water}) \] Here, the mass of the water \(m_{water} = 1.00 \ kg\), specific heat of water \(c_{water} = 4,180 \ J/(kg \cdot °C)\), and temperature changes need calculation.

Step 3: Calculate Temperature Changes

\(T_{initial, water} = 20.0^{\circ} \mathrm{C} \), \( T_{final} = 22.0^{\circ} \mathrm{C} \), and we assume \(T_{initial, metal} = 100.0^{\circ} \mathrm{C}\) primarily because it was in boiling water. Thus:\[ \Delta T_{water} = T_{final} - T_{initial, water} = 22.0^{\circ}C - 20.0^{\circ}C = 2.0^{\circ}C \]\[ \Delta T_{metal} = T_{final} - T_{initial, metal} = 22.0^{\circ}C - 100.0^{\circ}C = -78.0^{\circ}C \]

Step 4: Solve for the Specific Heat of the Metal

Insert the known values into the equation solved in Step 2:\[ m_{metal} = 0.5 \ kg \; m_{water} = 1.0 \ kg \; c_{water} = 4,180 \ J/(kg \cdot °C) \]Substitute these into:\[ 0.5 \cdot c_{metal} \cdot (-78.0) = -1.0 \cdot 4,180 \cdot 2.0 \]Solving this gives:\[ c_{metal} = \frac{8,360}{39} \approx 214.36 \ J/(kg \cdot °C) \]

Step 5: Compare Thermal Storage Capability

Water's specific heat is much higher \(\approx 4,180 \ J/(kg \cdot °C)\) compared to the metal's \(\approx 214.36 \ J/(kg \cdot °C)\), indicating water is more useful for storing thermal energy because it can absorb or release more heat per unit mass per degree of temperature change.

Step 6: Consider Styrofoam Heat Absorption

If the Styrofoam absorbed a non-negligible amount of heat, then the water would absorb less heat than calculated, leading to an overestimation of heat transferred by the metal. This would result in an overestimated specific heat of the metal being lower than the true value.

Key Concepts

Heat Exchange

Heat exchange refers to the process where thermal energy is transferred between two materials or substances. In the given exercise, this concept is crucial, as it describes the interaction between a hot metal chunk and cooler water. When the metal is placed in the water, it releases heat until both materials reach the same temperature.
This is called reaching thermal equilibrium.

• The metal loses heat because it was initially at a higher temperature.
• The water gains heat as it was initially cooler.

The principle of conservation of energy holds here, which means the heat lost by the metal is equal to the heat gained by the water.
Mathematically, this can be expressed as: \[ q_{metal} = -q_{water} \] where \( q \) represents the amount of heat, ensuring no heat is lost to the surroundings if we assume ideal conditions.

Thermal Energy Storage

Thermal energy storage is the ability of a material to absorb and retain heat. In general, materials with high specific heat capacities are effective for thermal energy storage.
In the exercise, we compare the thermal storage capacities of an unknown metal and water:

• Water has a specific heat capacity of about 4,180 J/(kg°C).
• The metal has a calculated specific heat of approximately 214.36 J/(kg°C).

This stark difference indicates that water can store more thermal energy per unit weight compared to the metal. This is why water is often used in heating systems and cooling applications, as it can buffer temperature changes effectively by absorbing significant amounts of heat without undergoing a large change in temperature. Conversely, the metal would warm up or cool down much quicker due to its lower specific heat capacity.

Heat Capacity

Heat capacity is a measure of a substance's ability to absorb heat for a given temperature change. It is an extensive property, meaning it depends on the amount of the substance present. Related to this is specific heat capacity, an intensive property that refers to how much heat is needed to change the temperature of a unit mass of a substance by one degree Celsius.
In the exercise, the specific heat capacity is critical for solving how much heat the metal and water exchange. The metal's specific heat was calculated to be 214.36 J/(kg°C), while for water, it is about 4,180 J/(kg°C).

• Higher specific heat means a material can "store" more heat.
• Materials with low specific heat heat up or cool down more quickly.

Thus, in terms of thermal management, knowing a material's heat capacity is crucial, as it directly influences how substances respond to thermal energy changes in applications like cooking, engineering, and climate control.

Temperature Change

Temperature change is the difference in temperature a material undergoes when it gains or loses thermal energy. In the case of this exercise, both the metal and the water underwent changes in temperature to reach equilibrium.

• The water's temperature increased from 20°C to 22°C.
• The metal's temperature decreased from 100°C to 22°C.

The magnitude of temperature change directly affects the amount of heat exchanged. For instance, a small temperature change in water required a significant heat exchange because of water's high specific heat capacity, compared to the metal which experienced a larger temperature drop to release the same amount of heat.
Understanding how temperature change interacts with material properties like specific heat allows for predicting thermal responses in energy systems and everyday contexts, making it a fundamental concept in thermodynamics.