Question

Suppose you have a 2.000-pound block of iron at \(50.0^{\circ} \mathrm{C}\). How much heat in joules would it take to warm this block to \(75.0^{\circ} \mathrm{C}\) ?

Short answer

The heat required to warm the iron block to \(75.0^{\circ}\mathrm{C}\) is approximately \(1.0 \times 10^4 \mathrm{J}\).

Step-by-step solution

Determine mass of the iron block (in kilograms)

Given the mass of the iron block is 2.000 pounds, we need to convert it into kilograms (kg) for our calculation since we'll be using SI units. The conversion factor is 1 pound = 0.453592 kg. Mass in kg = Mass in pounds × Conversion factor Mass in kg = 2.000 pounds × 0.453592 kg/pound Mass in kg = 0.907185 kg

Find the specific heat capacity of iron

The specific heat capacity of a substance is the amount of heat needed to raise the temperature of 1 kg of the substance by 1°C. To determine the heat required to warm the iron block, we need to know the specific heat capacity of iron. The specific heat capacity of iron is 449 J/kg·°C.

Calculate the temperature change

The initial temperature of the iron block is given as 50.0°C, and it needs to be heated to reach a final temperature of 75.0°C. We can find the temperature change by subtracting the initial temperature from the final temperature. ΔT = T_final - T_initial ΔT = 75.0°C - 50.0°C ΔT = 25.0°C

Use the formula to calculate heat required

Now that we have all the necessary values, we can use the formula for calculating heat absorbed or released during a change in temperature: Q = mcΔT Where Q represents the heat in joules, m is the mass of the iron block, c is the specific heat capacity of iron, and ΔT is the temperature change. Q = (0.907185 kg) × (449 J/kg·°C) × (25.0°C) Q = 10197.798 J

Round the answer to two significant figures

Since the least number of significant digits is 2 (the given mass is 2.000), we round the answer to two significant digits. The heat required to warm the iron block to 75.0°C is: Q = 1.0 × 10^4 J

Key concepts

Specific Heat Capacity

Specific heat capacity is an important concept in heat transfer, measuring the amount of energy needed to raise the temperature of 1 kilogram of a substance by 1 degree Celsius. It's specific because this value can vary widely between different materials. For example, water has a high specific heat capacity compared to metals, meaning it requires more energy to increase its temperature.
- For our exercise, we used the specific heat capacity of iron, which is 449 J/kg·°C. This tells us that for every kilogram of iron, 449 joules are needed to increase its temperature by 1°C.
Understanding this property helps in designing systems where heat management is critical, such as engines or cooking appliances.
By understanding a material’s specific heat capacity, we can predict how it will absorb energy and how quickly it will heat up or cool down.

Temperature Change

Temperature change, often notated as ΔT, reflects the difference between an initial and final temperature. It is a crucial parameter in calculating the heat absorbed or released by a substance.
- For our iron block, the initial temperature is 50°C and the target temperature is 75°C. Thus, the temperature change is calculated as:
\[ΔT = T_{final} - T_{initial} = 75°C - 50°C = 25°C\]This change indicates how much the iron’s temperature needs to increase.

• The larger the temperature change, the more energy is required for the heating process.
• This factor is directly proportional to the energy involved, which we see in the equation for heat transfer: \( Q = mcΔT \).

Understanding temperature change helps you grasp how varying conditions affect energy requirements.

Joules

Joules are the standard unit used to measure energy, specifically in the field of heat transfer. One joule is defined as the amount of energy transferred when a force of one newton is applied over a distance of one meter.
- In thermal calculations, joules tell us how much heat energy is required to change a substance's state or temperature.
In our exercise, the heat required to change the temperature of the iron block is calculated in joules. By applying the formula:\[Q = mcΔT\]we determined that 10197.798 joules are needed.

• This value helps in understanding the energy efficiency and requirements for heating materials.
• Converting joules into other units, like calories or kilojoules, is common for dietetics or chemistry contexts.

In science and engineering, quantifying heat transfer in joules allows clear communication and consistency across calculations.

Mass Conversion

Mass conversion is the process of converting the mass from one unit to another to ensure compatibility with scientific formulas, which commonly use SI units like kilograms.
- For the problem at hand, we needed to convert the iron block’s mass from pounds to kilograms.
Given:\[1 \, ext{pound} = 0.453592 \, ext{kilograms}\] The calculation converts the 2.000-pound mass of the iron to:\[Mass = 2.000 \, ext{pounds} \times 0.453592 \, ext{kg/pound} = 0.907185 \, ext{kg}\]Converting to kilograms makes calculations with specific heat capacity straightforward, as this unit is the standard in thermal calculations.

• Knowing mass conversion is essential for ensuring precise and accurate scientific calculations.
• Aids in reducing errors that occur when misaligned units are used.

Mastering unit conversions can help when dealing with both practical and theoretical physics problems involving mass.