Problem 142 - Introductory Chemistry Atoms First

Question

The specific heat of methane gas is \(2.20 \mathrm{~J} / \mathrm{g} \cdot{ }^{\circ} \mathrm{C}\). If the temperature of a sample of methane gas rises by \(15^{\circ} \mathrm{C}\) when \(8.8 \mathrm{~kJ}\) of heat energy is added to the sample, what is the mass of the sample?

Step-by-Step Solution

Answer

The mass of the methane gas sample is approximately \(266.67 \textrm{ grams}\) when 8.8 kJ of heat energy is added, and the temperature rises by \(15^{\circ} \mathrm{C}\).

Step 1: Convert given units

First, let's convert the given units to the standard units. The heat energy Q is given as 8.8 kJ, which needs to be converted to J (joules). 1 kJ = 1000 J So, 8.8 kJ = 8.8 × 1000 J = 8800 J

Step 2: Identify the given values

Now, let's identify the given values: Specific heat (c) = 2.20 J/g°C Temperature change (ΔT) = 15°C Heat energy (Q) = 8800 J

Step 3: Use the formula Q = mcΔT

Now we can use the formula for heat transfer to find the mass of the sample: Q = mcΔT Rearrange the formula to solve for the mass (m): m = Q / (cΔT)

Step 4: Plug in the given values and solve for mass

Now, plug in the given values to calculate the mass: m = (8800 J) / (2.20 J/g°C × 15°C) m = (8800 J) / (33 J/g) m ≈ 266.67 g

Step 5: State the result

The mass of the methane gas sample is approximately 266.67 grams.

Key Concepts

Specific Heat

Understanding specific heat is crucial when studying how substances absorb or release heat. Specific heat is defined as the amount of heat per unit mass required to raise the temperature of a substance by one degree Celsius. In essence, it describes how a particular material responds to heat input.
For instance, methane gas has a specific heat of 2.20 J/g°C, which means it takes 2.20 joules of energy to raise one gram of methane by one degree Celsius. This property is important because it allows us to predict how much energy is needed to change the temperature of a given mass of a substance.
In a mathematical expression, if we know the specific heat (c), the mass (m), and the change in temperature (ΔT), we can calculate the amount of heat transferred (Q) using the formula:

• Q = mcΔT

This formula plays a significant role in solving many heat transfer problems involving gases, liquids, and solids.

Methane Gas

Methane, a simple hydrocarbon with the chemical formula CH₄, is a colorless and odorless gas at room temperature. It is the main component of natural gas, widely used as a fuel source. Methane’s molecular structure consists of one carbon atom bonded to four hydrogen atoms, forming a tetrahedral shape.
Methane is significant in atmospheric chemistry because it is a potent greenhouse gas. However, in the context of heat capacity, methane is simply a subject of study for its thermodynamic properties.
In our problem, we deal with methane’s specific heat capacity, which tells us how much energy it takes to increase the temperature of a given mass of methane gas. This knowledge is essential for applications involving the heating or cooling of methane in industrial processes and environmental studies.

Heat Transfer

Heat transfer is the process by which thermal energy moves from a hotter object to a cooler one. There are three main modes of heat transfer: conduction, convection, and radiation. In the given exercise, we focus on the heat transfer to methane gas, usually involving conduction and convection.
In our case study, we see how the concept of heat transfer applies when 8.8 kJ, or 8800 joules, of energy is added to a sample of methane. This energy causes a rise in temperature by 15°C.
The relationship between heat transfer, specific heat, mass, and temperature change is expressed through the equation:

• Q = mcΔT

This equation allows us to calculate different variables, such as the mass of the sample, when other variables are known. Understanding how to manipulate this formula is key to solving practical heat problems.

Temperature Change

In physics, temperature change is the difference between the initial and final temperatures of a substance after heat transfer occurs. It's a critical factor in determining the amount of heat transferred to or from a substance.
In our exercise, methane gas experiences a temperature change (ΔT) of 15°C after absorbing 8800 J of energy. The temperature change helps us understand how much energy is involved in altering the thermal state of a substance.
Through the formula:

• Q = mcΔT

We can relate the heat added to a system with its temperature change, specific heat, and mass. This particular relationship is indispensable for scientists and engineers when designing systems that manage heat, ensuring efficiency in processes ranging from residential heating to large-scale industrial applications.