Question

A hot-water bottle contains \(725 \mathrm{~g}\) of water at \(65^{\circ} \mathrm{C}\). If the water cools to body temperature \(\left(37^{\circ} \mathrm{C}\right)\), how many kilocalories of heat could be transferred to sore muscles?

Short answer

20.3 kilocalories

Step-by-step solution

- Understand the Heat Transfer

To find the amount of heat transferred, use the formula: \[ Q = mc\triangle T \] where \( Q \) is the heat transferred in calories, \( m \) is the mass of the water, \( c \) is the specific heat capacity of water, and \( \triangle T \) is the temperature change.

- Identify Given Values

From the problem, we know the following values: \( m = 725 \) grams \( c = 1 \) calorie/gram \, ^\backslash circ C \( T_{\text{initial}} = 65 \)^\backslash circ C \( T_{\text{final}} = 37 \)^\backslash circ C

- Calculate the Temperature Change

Determine the change in temperature \[ \triangle T = T_{\text{initial}} - T_{\text{final}} = 65^\backslash circ C - 37^\backslash circ C = 28^\backslash circ C \]

- Apply the Heat Transfer Formula

Substitute the known values into the heat transfer formula \[ Q = mc\triangle T = 725 \, \text{g} \times 1 \, \text{cal/g} \, ^\backslash circ \text{C} \times 28^\backslash circ \text{C} = 20300 \, \text{calories} \]

- Convert Calories to Kilocalories

Since there are 1000 calories in a kilocalorie, convert the result \[ Q = 20300 \, \text{calories} \times \frac{1 \, \text{kcal}}{1000 \, \text{calories}} = 20.3 \, \text{kcal} \]

Key concepts

Specific Heat Capacity

Specific heat capacity is a property of a substance that indicates how much heat is required to change the temperature of one gram of the substance by one degree Celsius. For water, this value is quite high, at 1 calorie/gram °C. This means that water can store a large amount of heat energy. It’s why water is used in hot-water bottles to relieve sore muscles.

The specific heat capacity is essential in calculating heat transfer because it tells us how resistant the material is to changing temperature. Knowing the specific heat capacity helps us determine the amount of energy needed to warm up or cool down a specific mass of the substance. This concept is useful in many practical applications, like heating homes or cooking.

Temperature Change

Temperature change, denoted as \(\triangle T\), is the difference between the initial and final temperatures of a substance. In our exercise, the initial temperature \(T_i\) is 65°C, and the final temperature \(T_f\) is 37°C. The change in temperature is calculated as:
\[ \triangle T = T_{\text{initial}} - T_{\text{final}} = 65^\backslash circ \text{C} - 37^\backslash circ \text{C} = 28^\backslash circ \text{C} \]

Understanding temperature change helps us understand how much energy is needed to achieve a temperature difference. When a substance undergoes a temperature change, it either absorbs or releases energy. In the example, the water in the hot-water bottle cools down, losing energy which can potentially be used for therapeutic purposes.

Energy Conversion

Energy conversion involves changing energy from one form to another. In our context, we are primarily concerned with converting thermal energy stored in water to another form of energy, such as kinetic or mechanical energy for muscle relaxation.

In chemistry, we often measure energy in calories or kilocalories. One kilocalorie (kcal) is equivalent to 1000 calories, and it's crucial to perform this conversion when discussing larger amounts of energy. In our exercise, the heat transferred was calculated as 20300 calories. To make it more understandable, we convert it into kilocalories:
\[ Q = \frac{20300 \text{ calories}}{1000 \text{ calories/kcal}} = 20.3 \text{ kilocalories} \]

This process of converting units helps in practical applications, such as calculating dietary energy or understanding the energy expenditure in physical activities. Energy conversion is fundamental to grasp as it ties physical transformations to their measurable impacts in real-world scenarios.