Problem 12 - Chemistry Matter and Change

Question

A 90.0 -g sample of an unknown metal absorbed 25.6 \(\mathrm{J}\) of heat as its temperature increased \(1.18^{\circ} \mathrm{C}\) . What is the specific heat of the metal?

Step-by-Step Solution

Answer

The specific heat of the metal is approximately 0.241 J/g°C.

Step 1: Understand the problem

We are given a sample of metal with a mass of 90.0 g that absorbs heat, causing its temperature to rise by 1.18 °C. We need to find the specific heat capacity of the metal, which is a measure of how much heat energy is needed to change the temperature of 1 gram of that substance by 1 °C.

Step 2: Write the formula for specific heat

The formula to calculate specific heat capacity \(c\) is:\[ c = \frac{q}{m \cdot \Delta T} \]where \(q\) is the heat absorbed (in Joules), \(m\) is the mass of the substance (in grams), and \(\Delta T\) is the change in temperature (in °C).

Step 3: Plug in known values

Substitute the given values into the formula:- Heat absorbed, \(q = 25.6 \text{ J}\)- Mass, \(m = 90.0 \text{ g}\)- Temperature change, \(\Delta T = 1.18 \text{ °C}\)\[ c = \frac{25.6}{90.0 \times 1.18} \]

Step 4: Perform the calculation

Calculate the denominator first:\(90.0 \times 1.18 = 106.2\)Then, divide the heat energy by the result from the previous step:\[ c = \frac{25.6}{106.2} \approx 0.241 \text{ J/g°C} \]

Step 5: Conclude with final answer

The specific heat of the unknown metal is approximately 0.241 J/g°C. This means that 0.241 Joules of energy are needed to raise the temperature of 1 gram of the metal by 1 °C.

Key Concepts

Heat Absorption

Whenever a substance absorbs heat, its molecules become more energetic, moving faster and often leading to a temperature increase. In this exercise, a metal sample absorbs heat, which is measured in Joules (J). The metal takes in 25.6 J of heat, causing its temperature to rise.

• Heat absorption involves transferring energy into a substance, akin to charging a battery.
• This energy increase often results in a rise in temperature, but exactly how much depends on the material's properties, such as specific heat capacity.

Understanding how much heat a material can absorb without a significant temperature increase is vital, especially in materials science and engineering. It explains why some materials feel hotter to the touch than others, even if they're exposed to the same amount of heat.

Temperature Change

The change in temperature ( fT) of a sample is an important factor in calculating specific heat. In this problem, the metal's temperature rises by \(1.18^{\circ}\mathrm{C}\) due to heat absorption.

• Temperature change helps determine the relationship between absorbed heat and the material's thermal properties.
• A small temperature change relative to absorbed heat might indicate a high specific heat capacity, meaning the material resists temperature changes.

The relationship between heat and temperature change is not only crucial for chemistry but also for understanding everyday phenomena, like why metals heat up differently than water when exposed to sunlight.

Metal Sample

A metal sample serves as a model to study heat absorption and specific heat capacity. In this exercise, the metal sample is unknown, weighing 90.0 grams.

• Metals generally have varying specific heat capacities, which affect how quickly they warm up.
• Mass plays a role in such calculations, with larger masses needing proportionately more energy for the same temperature increase.

Studying how a specific metal behaves when heated can lead to discoveries in metallurgy, like which alloys to use for particular applications based on their thermal properties. By performing these calculations, we gain insights into the physical characteristics of the materials we use daily.

Energy Calculation

Calculating the energy needed to raise the temperature of a substance involves using the formula for specific heat. In this scenario, the calculation reveals how responsive the metal is to temperature changes.

• The specific heat capacity formula \(c = \frac{q}{m \cdot \Delta T}\) helps determine how much heat energy per gram per degree Celsius a substance can absorb.
• The result aids in identifying unknown materials by comparing calculated specific heat with known values.

By inputting the known variables—heat absorbed, mass, and temperature change—into the formula, we determine the specific heat capacity of the metal. Understanding this allows scientists and engineers to choose appropriate materials for thermal applications.