Question

What is the final temperature of a 3 kg wrought iron fireplace tool that is left in front of an electric heater, absorbing heat energy at a rate of \(100 \mathrm{W}\) for 10 minutes? Assume the pendant is initially at \(20^{\circ} \mathrm{C}\) and that the specific heat of wrought iron is \(233 \frac{\mathrm{J}}{\mathrm{kg} \cdot \mathrm{K}}\) a. \(40^{\circ} \mathrm{C}\) b. \(50^{\circ} \mathrm{C}\) c. \(60^{\circ} \mathrm{C}\) d. \(70^{\circ} \mathrm{C}\)

Short answer

The calculated final temperature is approximately 105.97°C, which does not match the provided options.

Step-by-step solution

- Identify the given data

We have the following data:- Mass of the wrought iron tool, m = 3 kg- Initial temperature, T_initial = 20°C- Heat energy absorption rate, P = 100 W- Time duration, t = 10 minutes (convert to seconds: 10 * 60 = 600 s)- Specific heat of wrought iron, c = 233 J/(kg·K)

- Calculate the total heat energy absorbed

The heat energy absorbed can be calculated using the formula: Q = P * t Substitute the given values: Q = 100 W * 600 s = 60000 J

- Calculate the temperature change

Use the formula for specific heat to find the temperature change: Q = m * c * ΔT Rearrange to solve for the temperature change, ΔT: ΔT = Q / (m * c) Substitute the known values: ΔT = 60000 J / (3 kg * 233 J/(kg·K))ΔT ≈ 85.97 K

- Calculate the final temperature

The final temperature is the initial temperature plus the temperature change: T_final = T_initial + ΔT Substitute the known values: T_final = 20°C + 85.97 K T_final ≈ 105.97°C

- Choose the correct option

The calculated final temperature of approximately 105.97°C does not match any provided options, suggesting there may be a misunderstanding in interpreting the question or an issue with the provided options. However, based on calculation steps, none of the provided options (a: 40°C, b: 50°C, c: 60°C, d: 70°C) match the calculated final temperature.

Key concepts

specific heat capacity

The specific heat capacity of a material is a measure of how much heat energy is required to raise the temperature of a given mass of the substance by one degree Celsius (or one Kelvin).
It is denoted by the letter 'c' in formulas and is usually expressed in units of joules per kilogram per degree Celsius (J/(kg·K)).
Different substances have different specific heat capacities, meaning they require different amounts of heat energy to achieve the same temperature change.
For example, water has a high specific heat capacity, which explains why it heats up and cools down more slowly compared to metals like iron, which have lower specific heat capacities.

heat energy absorption

Heat energy absorption refers to the process whereby an object takes in heat from its surroundings.
This can occur through various methods, such as conduction, convection, or radiation.
In the context of the exercise, the wrought iron tool absorbs heat from the electric heater via radiation.
We calculate the total heat energy absorbed using the power rating of the heater and the time duration of heating.
The formula used is:
\( Q = P \times t \)
where \( Q \) is the total heat energy absorbed, \( P \) is the power of the heater in watts, and \( t \) is the time in seconds.
In this problem, the heater has a power of 100 watts, and the tool is heated for 600 seconds (10 minutes converted to seconds), thus absorbing 60,000 joules of energy.

temperature change formula

To determine how much a substance's temperature will change when it absorbs a certain amount of heat energy, we use the temperature change formula.
This formula is derived from the relationship between heat energy, mass, specific heat capacity, and temperature change.
The formula is:
\( \)
Q = m \cdot c \cdot \Delta T \
where \( Q \) is the heat energy absorbed, \( m \) is the mass of the substance, \( c \) is the specific heat capacity, and \( \Delta T \) is the change in temperature.
Rearranging the formula to solve for the temperature change, \Delta T, we get: \
\( \Delta T = \frac{Q}{m \cdot c} \)
Plugging in our values from the exercise, with \( Q = 60000 \)J, \( m = 3 \)kg, and \( c = 233 \)J/(kg·K), we find:
\( \Delta T \approx 85.97 \)K.

heat transfer in physics

Heat transfer is a fundamental concept in physics that describes the movement of thermal energy from one object or substance to another.
This transfer takes place through three main mechanisms: conduction, convection, and radiation.
Conduction is the transfer of heat through direct physical contact, such as touching a hot stove.
Convection involves the transfer of heat through fluids (liquids or gases) due to the motion of the fluid itself.
Radiation does not require a medium and involves the transfer of heat through electromagnetic waves, such as the heat we feel from the sun.
In the exercise, the wrought iron fireplace tool received heat through radiation from the electric heater.
Understanding these methods of heat transfer helps us analyze and solve various problems relating to thermal energy in physics.