Problem 80 - Chemistry: Principles and Reactions

Question

Equal masses of liquid A, initially at \(100^{\circ} \mathrm{C}\), and liquid B, initially at \(50^{\circ} \mathrm{C}\), are combined in an insulated container. The final temperature of the mixture is \(80^{\circ} \mathrm{C}\). All the heat flow occurs between the two liquids. The two liquids do not react with each other. Is the specific heat of liquid \(A\) larger than, equal to, or smaller than the specific heat of liquid B?

Step-by-Step Solution

Answer

Answer: Liquid B has a larger specific heat.

Step 1: Write down the formula for heat transfer and the conservation of energy principle.

We can use the formula for heat transfer: \(q = mc\Delta T\), where \(q\) is the heat transferred, \(m\) is the mass, \(c\) is the specific heat, and \(\Delta T\) is the change in temperature. Since the container is insulated, the heat lost by liquid A must be equal to the heat gained by liquid B. Therefore, we have: \(q_A = q_B\).

Step 2: Calculate the heat transfers for both liquids A and B.

For liquid A, the temperature change is: \(\Delta T_A = T_{final} - T_{initial} = 80^{\circ} \mathrm{C} - 100^{\circ} \mathrm{C} = -20^{\circ}\mathrm{C}\) For liquid B, the temperature change is: \(\Delta T_B = T_{final} - T_{initial} = 80^{\circ} \mathrm{C} - 50^{\circ} \mathrm{C} = 30^{\circ}\mathrm{C}\)

Step 3: Use the conservation of energy equation to compare the specific heats.

Using the conservation of energy equation \(q_A = q_B\), we can write: \(m_A c_A \Delta T_A = m_B c_B \Delta T_B\) Since the masses of both liquids are equal, \(m_A = m_B\). We can cancel out the masses in the equation, resulting in: \(c_A \Delta T_A = c_B \Delta T_B\) Now we can plug in the temperature changes we calculated in Step 2: \(c_A(-20^{\circ}\mathrm{C}) = c_B(30^{\circ}\mathrm{C})\) From this equation, if \(c_A\) is larger than \(c_B\), the negative term on the left side would overpower the positive term on the right side, which is not possible as both sides must be equal. Therefore, the specific heat of liquid A must be smaller than the specific heat of liquid B. Answer: The specific heat of liquid A is smaller than the specific heat of liquid B.

Key Concepts

Heat Transfer

Heat transfer is a fundamental physical process where thermal energy moves from one body or substance to another. In our daily lives and in various scientific applications, understanding heat transfer is crucial.

There are three primary modes of heat transfer: conduction, convection, and radiation. Conduction occurs when heat flows through a solid or between objects in direct contact, convection involves the movement of fluid, such as air or water, which carries heat with it, and radiation is the transfer of heat through electromagnetic waves, such as when the sun heats the Earth.

In the context of our exercise, where two liquids of different temperatures are mixed in an insulated container, we are mainly concerned with conduction. No heat is lost to the surroundings due to the insulation, which simplifies our analysis. Heat moves from the hotter liquid (A) to the cooler liquid (B) until thermal equilibrium is reached; that is, until both liquids are at the same temperature.

Applying Heat Transfer to the Exercise

The formula for heat transfer, q = mctDelta T, allows us to calculate the quantity of heat (q) transferred between bodies based on their mass (m), specific heat (cc), and the change in temperature (Delta T). The principle of conservation of energy tells us that the heat lost by liquid A will be equal to the heat gained by liquid B, leading to the equation q_A = q_B}. It facilitates our understanding of how energy balance plays a role in predicting the final temperature of the mixture.

Conservation of Energy

Conservation of energy is a fundamental principle in physics which states that energy cannot be created or destroyed, only transferred or converted from one form to another. In an isolated system, the total energy remains constant over time. This principle is key to solving many problems in physics and engineering, including those involving heat transfer.

In our exercise, the principle of conservation of energy is explicitly used. The system comprises two liquids in an insulated container, which means it is isolated from its environment. The heat lost by the warmer liquid must equal the heat gained by the cooler liquid as they reach a mutual final temperature.

Putting Energy Conservation to Work

When we set the equation q_A = q_B up, we assume the conservation of energy within our system. By equating the heat lost and gained and knowing the masses are equal, we can cancel them out and compare the specific heats directly. This direct comparison allows us to conclude about the relative magnitudes of the specific heats of the two liquids, thus applying energy conservation to reach a conclusion on specific heat capacities.

Temperature Change

Temperature change is an essential concept in thermodynamics that indicates the difference in temperature as a system undergoes a process. It's the driver behind heat transfer; heat flows from a body of higher temperature to one of lower temperature until both achieve equilibrium.

In the context of our textbook exercise, the final temperature of the mixture allows us to calculate the temperature change for both liquids, which is essential for determining the direction and amount of heat flow. The temperature change is represented mathematically as Delta T = T_{final} - T_{initial}.

Analyzing Temperature Change

The negative temperature change for liquid A indicates it has released heat, while the positive change for liquid B indicates it has absorbed heat. By quantifying these changes, we can utilize the formula for heat transfer to find the relationship between the specific heats of the liquids. Since the final temperature of the mix is closer to the initial temperature of liquid A, it suggests that liquid A has a smaller specific heat capacity. The ability to analyze temperature changes is thus fundamental in solving problems related to heat transfer and specific heat.