Question

A 275 -g sample of a metal requires \(10.75 \mathrm{~kJ}\) to change its temperature from \(21.2^{\circ} \mathrm{C}\) to its melting temperature, \(327.5^{\circ} \mathrm{C}\). What is the specific heat of this metal?

Short answer

The specific heat of the metal is approximately 0.1276 J/(g°C).

Step-by-step solution

Identify Given Information

The mass of the metal sample is given as 275 g, the amount of heat transferred is 10.75 kJ, the initial temperature is 21.2°C, and the final temperature is 327.5°C. Note that we need to convert the heat transferred from kJ to J by multiplying by 1000.

Convert the Heat Energy to Joules

Since the specific heat capacity formula uses Joules, convert the given energy from kJ to J: Energy in Joules (Q) = 10.75 kJ * 1000 = 10750 J.

Calculate the Change in Temperature

The change in temperature (delta T) is the final temperature minus the initial temperature: Delta T = 327.5°C - 21.2°C = 306.3°C.

Apply the Formula for Specific Heat

Use the formula for specific heat capacity (c): Q = mc(delta T), where Q is the heat energy in Joules, m is the mass in grams, c is the specific heat capacity, and (delta T) is the change in temperature. Solve for c: c = Q / (m * delta T).

Calculate the Specific Heat Capacity

Substitute the given values into the formula and solve for c: c = 10750 J / (275 g * 306.3°C) = 10750 J / (84232.5 g°C) = approximately 0.1276 J/(g°C).

Key concepts

Heat Transfer

Heat transfer is a cornerstone of thermodynamics that describes the flow of thermal energy due to a temperature difference. Specifically, it occurs from a hotter object to a cooler object until equilibrium is reached. There are three main modes of heat transfer: conduction, which occurs through direct contact; convection, which involves the movement of fluid; and radiation, which transfers energy via electromagnetic waves.

In the context of calculating the specific heat capacity of a metal, heat transfer is directly related to the amount of energy, measured in joules, that the metal must absorb to increase its temperature. In our exercise, a metal sample absorbs 10.75 kJ of energy to increase its temperature from 21.2°C to its melting point at 327.5°C. Understanding heat transfer allows us to determine the amount of heat absorbed or released by the substance when undergoing a temperature change.

Temperature Change

Temperature change refers to the difference in temperature that a substance experiences when exposed to heat transfer. It is a measure of how much the thermal energy of the substance has increased or decreased. The concept of temperature change is essential when calculating specific heat capacity because it determines the amount of heat required to change the temperature of a material by a certain amount.

In our exercise, the temperature change (\( \text{Delta T} \) ) is calculated by subtracting the initial temperature from the final temperature. This change, which is 306.3°C for the metal sample, is used in the specific heat formula to compute the energy needed per gram per degree Celsius (\( \frac{J}{g°C} \) ). Recognizing this change is crucial to comprehend how temperature affects matter and to predict how substances will behave in thermal processes.

Thermodynamics

Thermodynamics is the branch of physics that deals with heat and temperature and their relation to energy and work. It is governed by the laws of thermodynamics, which describe how thermal energy is transferred and transformed in systems.

In this exercise, the first law of thermodynamics, also known as the law of conservation of energy, applies as the heat energy transferred to the metal is used to increase its temperature. No energy is lost or created in the process; it is merely transformed from one form to another. By calculating the specific heat capacity of the metal, we apply thermodynamic principles to identify how efficiently a substance can store thermal energy. This specific heat capacity is a fundamental property that enables scientists and engineers to design systems for heating and cooling, such as radiators, engines, or even cookware, by understanding how different materials react to heat.