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The specific heat of alcohol is about half that of water. Suppose you have \(0.5 \mathrm{kg}\) of alcohol at the temperature \(20^{\circ} \mathrm{C}\) in one container, and \(0.5 \mathrm{kg}\) of water at the temperature \(30^{\circ} \mathrm{C}\) in a second container. When these fluids are poured into the same container and allowed to come to thermal equilibrium, (a) is the final temperature greater than, less than, or equal to \(25^{\circ} \mathrm{C} ?\) (b) Choose the best explanation from among the following: I. The low specific heat of alcohol pulls in more heat, giving a final temperature that is less than \(25^{\circ}\). II. More heat is required to change the temperature of water than to change the temperature of alcohol. Therefore, the final temperature will be greater than \(25^{\circ}\). III. Equal masses are mixed together; therefore, the final temperature will be \(25^{\circ},\) the average of the two initial temperatures.

Step by step solution

01

Understand Specific Heat

The specific heat (\(c\)) of a substance is the amount of heat energy required to raise the temperature of a unit mass of the substance by one degree Celsius. For this problem, the specific heat of alcohol is half that of water. This implies \( c_{\text{alcohol}} = \frac{1}{2}c_{\text{water}} \).

02

Consider Heat Exchange

During thermal equilibrium, heat lost by water equals heat gained by alcohol because energy is conserved. The heat change (\(Q\)) is given by \(Q = mc\Delta T\). Let's consider the quantity of heat exchanged when they reach a common temperature \(T_f\).

03

Set Up the Heat Equation

Let \( m = 0.5 \, \text{kg} \) be the mass of both alcohol and water. The heat lost by water is \( Q_{\text{water}} = m c_{\text{water}} (30 - T_f) \), and the heat gained by alcohol is \( Q_{\text{alcohol}} = m \times \frac{1}{2}c_{\text{water}} (T_f - 20) \). At equilibrium, \[ mc_{\text{water}} (30 - T_f) = m \times \frac{1}{2}c_{\text{water}} (T_f - 20). \]

04

Simplify and Solve the Equation

Cancel out the common terms, \( m \) and \( c_{\text{water}} \), to get: \[ 30 - T_f = \frac{1}{2} (T_f - 20). \] Simplify and solve for \( T_f \): \[ 30 - T_f = \frac{1}{2}T_f - 10. \] Multiply the entire equation by 2 to eliminate the fraction: \[ 60 - 2T_f = T_f - 20. \] Rearrange to: \[ 3T_f = 80, \] therefore: \[ T_f = \frac{80}{3} \approx 26.7^{\circ}C. \]

05

Evaluate Options

The final temperature \( T_f \approx 26.7^{\circ}C \), which is greater than \( 25^{\circ}C \). The explanation II is correct: More heat is required to change the temperature of water than alcohol, resulting in a final temperature greater than \( 25^{\circ}C \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Specific Heat

Specific heat is an essential concept when studying thermal dynamics. It refers to the amount of heat energy required to raise the temperature of a unit mass of a substance by one degree Celsius. Specific heat is a unique property of each substance and helps us understand how different materials react to heat.
To calculate heat required or released by a substance, we use the formula: \[ Q = mc\Delta T \]where:

• \( Q \) is the heat energy transferred,
• \( m \) is the mass of the substance,
• \( c \) is the specific heat capacity,
• \( \Delta T \) is the change in temperature.

In our exercise, alcohol has about half the specific heat of water, meaning it requires less heat to change its temperature. This characteristic impacts how substances exchange heat, especially when mixed, as seen in the problem with alcohol and water.

Heat Exchange

When two substances at different temperatures are mixed, they exchange heat until reaching thermal equilibrium. This process is called heat exchange. In thermal equilibrium, the heat lost by the warmer substance equals the heat gained by the cooler substance. The principle of heat exchange is based on the conservation of energy.

• The hotter substance transfers its heat to the cooler one, cooling itself down.
• The cooler substance absorbs heat, warming up.

Each substance's ability to change temperature due to heat exchange heavily depends on its mass and specific heat. In the case of water and alcohol, given their sizes and properties, their final temperature when combined can be calculated using equations involving their specific heat capacities. This explains why water, which has a higher specific heat, loses heat slowly, impacting the final temperature after mixing.

Conservation of Energy

The cornerstone of understanding thermal equilibrium is the principle of conservation of energy. During heat exchange, energy is conserved; it is neither created nor destroyed, only transferred.
In the alcohol and water problem, we assume no external heat loss, meaning the combined system's total heat energy remains constant. When these substances are mixed:

• The heat lost by water equals the heat gained by alcohol.
• This relationship can be represented as:

\[ mc_{\text{water}} (T_{i_{\text{water}}} - T_f) = mc_{\text{alcohol}} (T_f - T_{i_{\text{alcohol}}}) \]where \( T_f \) is the final temperature, and \( T_{i_{\text{water}}} \) and \( T_{i_{\text{alcohol}}} \) are the initial temperatures of water and alcohol, respectively.
This equation simplifies and helps us find the equilibrium temperature, showcasing how energy conservation allows us to predict the outcome of heat exchanges accurately. Hence, the final temperature is determined by both substances' initial states and their specific thermal properties.