Question

If \(69.5 \mathrm{~kJ}\) of heat is applied to a 1012 -g block of metal, the temperature of the metal increases by 11.4 'C Calculate the specific heat capacity of the metal in \(\mathrm{J} / \mathrm{g} \quad \mathrm{C}\)

Short answer

The specific heat capacity of the metal is approximately \(6.02 \dfrac{\mathrm{J}}{\mathrm{g} \cdot \mathrm{°C}}\).

Step-by-step solution

Rearrange the formula

We need to rearrange the formula to isolate the specific heat capacity, \(c\). To do this, we can divide both sides of the equation by \(m\) and \(\Delta T\): \(c = \dfrac{q}{m\Delta T}\)

Use the given values

Now we can plug in the given values in the problem: - \(q = 69.5 \mathrm{~kJ}\) (heat applied, which we need to convert to Joules, since we need the specific heat in J/g°C), - \(m = 1012 \mathrm{~g} \) (mass of the metal block), - \(\Delta T = 11.4 \mathrm{~°C}\) (increase in temperature).

Convert the heat applied from kJ to J

To convert \(q\) from kJ to J, we need to multiply by 1000: \(q = 69.5 \mathrm{~kJ} \times 1000 = 69500 \mathrm{~J}\)

Calculate the specific heat capacity

Now we can substitute the converted value of \(q\) and other given values into the formula we derived in step 1: \(c = \dfrac{69500 \mathrm{~J}}{1012 \mathrm{~g} \times 11.4 \mathrm{~°C}} = \dfrac{69500}{11536.8}\) \(c \approx 6.02 \dfrac{\mathrm{J}}{\mathrm{g} \cdot \mathrm{°C}}\)

Present the final answer

The specific heat capacity of the metal is approximately \(6.02 \dfrac{\mathrm{J}}{\mathrm{g} \cdot \mathrm{°C}}\).

Key concepts

Heat Transfer

When studying heat transfer, it is essential to understand that it refers to the movement of thermal energy from one object or substance to another. This process can occur in three main ways: conduction, convection, and radiation. In the context of the given exercise, the object receiving the heat is a metal block. Specifically, a quantity of energy measured at 69.5 kJ is being applied to it.

• Conduction: This is how heat is transferred within the metal block. When heat is applied, it moves through the metal as the molecules vibrate and transfer energy to neighboring molecules.
• Convection: Though not directly involved in this exercise, convection occurs in fluids (liquids or gases) where hot parts of the fluid move and carry heat with them.
• Radiation: This involves heat being transferred in the form of electromagnetic waves and is less relevant here since the heat source is in direct contact with the metal.

This heat transfer causes the object—in our case, the metal bar—to increase its temperature, allowing us to calculate specific heat capacity.

Temperature Change

Temperature change is a crucial part of this exercise, as it directly influences how we calculate specific heat capacity. The change in temperature signifies how much the material's internal energy has altered due to heat transfer. In this scenario, we observed an increase in the temperature of the metal block by 11.4°C.

When calculating the specific heat capacity, the change in temperature (\( \Delta T \)) needs to be considered as follows:

• Start with the initial temperature of the metal (considered before applying heat).
• Subtract this from the final temperature to find \( \Delta T \).

This difference tells us how the material responded to the heat energy applied, giving insights into its thermal properties. Understanding this can help us figure out how different materials react to heat, an important factor in material science and engineering.

Energy Conversion

Energy conversion plays a pivotal role in understanding how heat energy is used and measured. In this exercise, we have an energy quantity initially given in kilojoules (kJ), where energy needs to be converted to a different measurement unit for further calculations. Specifically, we converted the energy from kJ to Joules (J) since specific heat capacity is expressed in J/g°C.

• Conversion steps involve using a simple multipicative factor (1 kJ = 1000 J).
• This ensures consistency in units used for measuring specific heat capacity.

These conversions are crucial in calculations involving energy-related concepts, allowing us to accurately work with various metric units seamlessly. By understanding these conversions, students can apply the knowledge practically, understanding how numerical values in one unit can be effectively utilized in different contexts.