Problem 45 - Physics

Question

At a fabrication plant, a hot metal forging has a mass of \(75 \mathrm{~kg}\) and a specific heat capacity of \(430 \mathrm{~J} /\left(\mathrm{kg} \cdot \mathrm{C}^{\circ}\right) .\) To harden it, the forging is immersed in \(710 \mathrm{~kg}\) of oil that has a temperature of \(32^{\circ} \mathrm{C}\) and a specific heat capacity of \(2700 \mathrm{~J} /\left(\mathrm{kg} \dot{\mathrm{c}} \mathrm{C}^{\circ}\right) .\) The final temperature of the oil and forging at thermal equilibrium is \(47^{\circ} \mathrm{C}\). Assuming that heat flows only between the forging and the oil, determine the initial temperature of the forging.

Step-by-Step Solution

Answer

The initial temperature of the forging is 939°C.

Step 1: Understanding Heat Transfer

When two objects are in thermal contact, heat flows from the hotter object to the cooler one until they reach the same temperature (thermal equilibrium). Here, heat lost by the metal forging will be equal to the heat gained by the oil.

Step 2: Using the Heat Transfer Formula

The formula for heat transfer is given by: \[ Q = m imes c imes (T_{final} - T_{initial}) \] where \( Q \) is the heat transferred, \( m \) is the mass, \( c \) is the specific heat capacity, and \( T \) is the temperature.

Step 3: Writing Equations for Heat Lost and Gained

Let \( T_f \) be the final temperature and \( T_i \) the initial temperature of the metal. The heat lost by the metal (\( Q_{metal} \)) is given by: \[ Q_{metal} = 75 imes 430 imes (47 - T_i) \]The heat gained by the oil (\( Q_{oil} \)) is given by:\[ Q_{oil} = 710 imes 2700 imes (47 - 32) \]

Step 4: Setting Heat Lost Equal to Heat Gained

According to the principle of conservation of energy, the heat lost by the metal forging equals the heat gained by the oil:\[ 75 imes 430 imes (47 - T_i) = 710 imes 2700 imes (47 - 32) \]

Step 5: Solving for Initial Temperature of the Forging

First, calculate the heat gained by the oil:\[ Q_{oil} = 710 imes 2700 imes 15 = 28,755,000 \] Substitute back into the equation for heat lost by the metal:\[ 75 imes 430 imes (47 - T_i) = 28,755,000 \]Solve for \( T_i \):\[ (47 - T_i) = \frac{28,755,000}{75 imes 430} \]\[ 47 - T_i = 892 \]\[ T_i = 47 - 892 \]\[ T_i = 939 \text{ °C} \]

Step 6: Conclusion

The initial temperature of the forging is 939°C.

Key Concepts

Specific Heat Capacity

Specific heat capacity is a fundamental concept in understanding heat transfer. It tells us how much heat energy is needed to change the temperature of a given amount of substance. It is usually denoted as \( c \) and is expressed in units of Joules per kilogram per degree Celsius \((J/(kg \cdot °C))\).
Each material has its own unique specific heat capacity, acting like a thermal fingerprint. For example:

• Materials with high specific heat capacity need more energy to change temperature.
• Water has a high specific heat capacity, which is why it heats up and cools down slowly.

In the exercise, the metal forging and oil have different specific heat capacities, which affect how they exchange heat. The metal gets much hotter quicker than the oil because it has a lower specific heat capacity. This difference is crucial in calculating the initial temperature of an object using the formula for heat transfer: \[ Q = m \cdot c \cdot (T_{\text{final}} - T_{\text{initial}}) \].
Understanding this helps us solve problems involving thermal equilibrium and energy exchange.

Thermal Equilibrium

Thermal equilibrium occurs when two substances at different temperatures reach a common temperature after being placed in contact. Energy in the form of heat flows from the hotter substance to the cooler one until their temperatures converge.
In the exercise, thermal equilibrium is achieved when the metal forging and the oil reach the same temperature, which is given as 47°C. When these two substances reach thermal equilibrium:

• The heat lost by the hotter object (metal) equals the heat gained by the cooler object (oil).
• The total energy of the system remains constant due to the conservation of energy principle.

Reaching thermal equilibrium allows us to use the final temperature to set up equations and find unknowns, such as the initial temperature of the forging using heat transfer calculations. This concept helps us comprehend how energy redistribution works in practical thermal situations.

Conservation of Energy

Conservation of energy is a central idea in physics, stating that energy in a closed system is constant. It cannot be created or destroyed, only transferred or transformed.
This principle is vital in understanding heat transfer between objects. In the context of the exercise, when the hot metal forging is immersed into oil:

• The energy lost by the metal is equal to the energy gained by the oil.
• The equation that adheres to conservation of energy is \( Q_{\text{lost}} = Q_{\text{gained}} \).

This leads us to the energy balance equation used in the solution:\[ 75 \times 430 \times (47 - T_i) = 710 \times 2700 \times (47 - 32) \].
Through this equation, we calculate the initial temperature of the metal forging. Acknowledging the conservation of energy ensures accurate predictions of how energy shifts in systems where heat transfer is involved.