Question

A 90.0-g sample of an unknown metal absorbed 25.6 J of heat as its temperature increased 1.18°C. What is the specific heat of the metal?

Short answer

The specific heat capacity of the unknown metal is approximately 0.243 J/g·°C.

Step-by-step solution

Identify the given values

In this problem, we have these given values: - Mass of the metal (m) = 90.0 g - Heat absorbed (q) = 25.6 J - Change in temperature (ΔT) = 1.18°C

Use the heat transfer equation

To find the specific heat capacity of the metal, we use the heat transfer equation: q = mcΔT Where: - q is the heat (in Joules) - m is the mass (in grams) - c is the specific heat capacity (in J/g·°C) - ΔT is the temperature change (in °C)

Rearrange the equation to solve for specific heat capacity (c)

We want to find c, the specific heat capacity. So we rearrange the equation like this: c = q / (mΔT)

Plug the given values into the equation

Now we can plug the given values into the equation: c = 25.6 J / (90.0 g * 1.18°C) Now, we will perform the calculation:

Perform the calculation

Using the values given, calculate the specific heat capacity: c = 25.6 J / (90.0 g * 1.18°C) c ≈ 0.243 J/g·°C

State the final answer

The specific heat capacity of the unknown metal is approximately 0.243 J/g·°C.

Key concepts

heat transfer equation

The heat transfer equation is a fundamental concept in physics that describes how heat energy is transferred between substances. In the exercise you encountered, we dealt with the equation:
\[ q = mc\Delta T \]
This equation is used to calculate the amount of heat absorbed or released by a substance when its temperature changes. Here's what each part of the equation means:

• q: Represents the amount of heat energy, measured in Joules (J).
• m: The mass of the substance, measured in grams (g).
• c: Specific heat capacity, which indicates how much heat is needed to raise the temperature of 1 gram of the substance by 1°C, measured in J/g·°C.
• \( \Delta T \): The change in temperature, calculated as the final temperature minus the initial temperature, measured in °C.

Using this equation allows us to understand how much heat is involved in changing the temperature of a substance. By rearranging the equation, you can solve for different variables depending on the information you have. In the original exercise, you rearranged it to find the specific heat capacity (c).
Although it might seem complex at first, with practice, using the heat transfer equation becomes a straightforward way to solve problems related to energy changes in materials.

calorimetry

Calorimetry is the science of measuring heat changes during a chemical reaction or physical process. It helps us understand how energy is transferred within a system. In the context of the exercise, it's important to know that we are working with a calorimeter setup that allows us to measure the heat changes associated with the metal sample.

To perform calorimetry, you would typically use a calorimeter, which isolates the system to ensure accurate measurement of heat transfer. However, in simplified problems like the one we've solved, we determine heat changes theoretically using the mentioned heat transfer equation.
Here, the metal sample absorbs heat energy, which causes a rise in its temperature. The key goals of calorimetry are:

• Measuring the heat absorbed or released in a process.
• Determining the specific heat capacity of substances, which tells us how they react to heat changes.
• Understanding the flow of energy in and out of the system.

While the original problem does not include the experimental setup, it's crucial to appreciate the principle of calorimetry to understand how heat energy is quantified and analyzed in practical scenarios.

temperature change

Temperature change (\( \Delta T \)) is a key part of understanding energy transfer in materials that undergo heating or cooling. It represents the difference in temperature as a material gains or loses heat. Specifically, for the equation we derived earlier, \( \Delta T \) is calculated as:
\[ \Delta T = T_{final} - T_{initial} \]
In the original exercise, you were given the temperature change directly as 1.18°C. This means that the metal's temperature increased by 1.18°C when it absorbed the heat.

Why Is Temperature Change Important?

It helps quantify the exact thermal response of a material when energy is added or removed. Recognizing the temperature change is critical in applying the heat transfer equation accurately because it directly affects the amount of heat required to produce that change.

Factors Affecting Temperature Change

• Specific Heat Capacity: Materials with higher specific heat capacities will require more energy to achieve the same temperature change compared to those with lower specific heat capacities.
• Mass of the Substance: The larger the mass, the more energy is needed to alter its temperature by a given amount.

Understanding these elements makes it easier to analyze and predict how substances respond under different thermal conditions.