Problem 164 - Introductory Chemistry Atoms First

Question

If the same amount of heat energy is added to a beaker containing \(100 \mathrm{~mL}\) of ethanol and a beaker containing \(100 \mathrm{~mL}\) of water, which liquid experiences the greater rise in temperature?

Step-by-Step Solution

Answer

When the same amount of heat energy is added to 100 mL of ethanol and 100 mL of water, ethanol will experience the greater rise in temperature due to its smaller specific heat capacity and lower mass.

Step 1: The formula to calculate the heat energy (Q) added to or removed from a substance is: \[Q = mc\Delta T\] where: - Q is the heat energy (in joules) - m is the mass of the substance (in grams) - c is the specific heat capacity of the substance (in J/g°C) - ΔT is the change in temperature (in °C) #Step 2: Plug in given values and solve for the change in temperature#

For this problem, we are told that: - The volume of each liquid is 100 mL. We can assume that the density of water is 1 g/mL, and the density of ethanol is 0.789 g/mL. Hence, the mass of water is 100 g, and the mass of ethanol is 78.9 g. - The specific heat capacity of water is 4.18 J/g°C and that of ethanol is 2.44 J/g°C. - The same amount of heat energy (Q) is added to both liquids. First, we'll solve for ΔT for water: Q = (100 g)(4.18 J/g°C)(ΔT_W) ΔT_W = Q / (100 g × 4.18 J/g°C) Now, we'll solve for ΔT for ethanol: Q = (78.9 g)(2.44 J/g°C)(ΔT_E) ΔT_E = Q / (78.9 g × 2.44 J/g°C) #Step 3: Compare temperature changes#

Step 2: We can see that: ΔT_W = (Q / (100 g × 4.18 J/g°C)) and ΔT_E = (Q / (78.9 g × 2.44 J/g°C)) Since the denominator in the ΔT_E equation is smaller than that in the ΔT_W equation, the value of ΔT_E will be greater than the value of ΔT_W. Therefore, ethanol will experience a greater rise in temperature. #Conclusion#

When the same amount of heat energy is added to 100 mL of ethanol and 100 mL of water, ethanol will experience the greater rise in temperature due to its smaller specific heat capacity and lower mass.

Key Concepts

Understanding Specific Heat Capacity

Specific heat capacity is a fundamental concept in chemistry that describes how much heat energy is required to raise the temperature of a unit mass of a substance by one degree celsius. It's essential to remember that every substance has a different capacity to store heat; for instance, water has a high specific heat capacity, which means it takes more energy to increase its temperature compared to other substances.

The specific heat capacity can be expressed with the formula:
\[\begin{equation}c = \frac{Q}{m\triangle T}\end{equation}\]
where:

• c represents the specific heat capacity,
• Q is the heat energy in joules,
• m is the mass of the substance in grams, and
• \triangle T is the temperature change in degrees Celsius.

Considering the specific heat capacity allows us to predict how a substance will react to the introduction of heat and explains why certain materials heat up or cool down more quickly than others. This property is a critical consideration in a wide array of practical applications, from culinary arts to engineering.

Calculating Temperature Change

When it comes to understanding how a substance reacts to added heat, calculating the temperature change is a pivotal step. The formula to determine the amount of heat energy absorbed or released by a substance is given as:
\[\begin{equation}Q = mc\triangle T\end{equation}\]
As established,

• Q is the heat energy,
• m is the mass,
• c is the specific heat capacity, and
• \triangle T is the temperature change.

Through rearranging the formula, we can solve for the variable \triangle T to find the change in temperature as a result of heat transfer:
\[\begin{equation}\triangle T = \frac{Q}{mc}\end{equation}\]
This equation illustrates the direct relationship between the heat energy and the resulting temperature change, accounting for the specific heat capacity and the mass of a substance. Understanding how to manipulate this relationship is crucial for scientists and engineers when designing systems for heating or cooling substances, controlling reactions, or managing energy resources.

Heat Transfer in Substances

Heat transfer is the process that occurs when heat energy is added to or removed from a substance, causing a change in temperature. The manner in which this heat is transferred plays a significant role in the subsequent temperature change. Common modes of heat transfer include conduction, convection, and radiation. In many chemistry problems, though, we focus mainly on the energy transfer through conduction, as substances come into contact, or specific heat transfer, which is where the concept of specific heat capacity is critical.

In the context of our exercise, when comparing the temperature change between ethanol and water, we see that ethanol, with its lower specific heat capacity, requires less energy to increase its temperature. This leads to a more significant temperature rise for a given amount of heat energy, as the heat added is not spread out over as great a capacity for absorption. This property of ethanol is what makes it more susceptible to temperature changes compared to water, which has one of the highest specific heat capacities of any substance. Therefore, understanding how heat transfer works in different substances helps in predicting outcomes in practical situations, from everyday cooking to industrial processes.