Question

A \(2.00-\mathrm{g}\) peanut is burned in a bomb calorimeter containing \(1200 \mathrm{~g}\) of water. The temperature of the water increases from \(25.00^{\circ} \mathrm{C}\) to \(30.25^{\circ} \mathrm{C}\). (a) How much heat, in joules, did the peanut release as it burned? (b) Calculate the heat content in units of calories and Calories. (c) Calculate the energy value in units of \(\mathrm{Cal} / \mathrm{g}\).

Short answer

a) The peanut released 26328 Joules. b) The heat content is 6294 calories and 6.294 Calories. c) The energy value is 3.147 Calories/gram.

Step-by-step solution

Calculate the heat released in joules

The thermal energy released by the peanut and absorbed by the water can be calculated using the formula of the specific heat capacity: \( Q = mc\Delta T \), where \( Q \) is the heat, \( m \) is the mass of the water, \( c \) is the specific heat capacity of water, and \( \Delta T \) is the temperature change. The specific heat capacity of water is \( 4.184 \, \mathrm{J / g \cdot K} \), the temperature change \( \Delta T \) is \( (30.25 - 25.00) \, ^\circ C = 5.25 \, ^\circ C \), and the mass of water is \( 1200 \, g \). When substituting these values, we get \( Q = 1200 \, g \times 4.184 \, \mathrm{J / g \cdot K} \times 5.25 \, K = 26328 \, J \).

Convert the heat in calories

The energy can be converted from joules to calories using the conversion factor \( 4.184 \, J = 1 \, cal \). Hence, the energy content in calories is \( 26328 \, J \times \left( \frac{1 \, cal}{4.184 \, J} \right) = 6294 \, cal \).

Convert the heat in Calories

The energy content can be converted from calories to Calories using the relation \( 1 \, Cal = 1000 \, cal \). Hence, the energy in Calories is \( 6294 \, cal \times \left( \frac{1 \, Cal}{1000 \, cal} \right) = 6.294 \, Cal \).

Compute the energy value in \(\mathrm{Cal / g}\)

The energy value in \(\mathrm{Cal / g}\) can be calculated by dividing the energy in Calories by the mass of the peanut. Thus, the energy value is \( 6.294 \, Cal \div 2.00 \, g = 3.147 \, \mathrm{Cal / g} \).

Key concepts

Understanding Specific Heat Capacity

Specific heat capacity is a key concept that helps us understand how different substances absorb or release heat. It tells us how much heat is needed to change the temperature of 1 gram of a substance by 1 degree Celsius. For example, water has a specific heat capacity of 4.184 J/g°C. This means it takes 4.184 joules of energy to increase the temperature of 1 gram of water by 1°C.

In the exercise, we used this to find out how much heat was absorbed by the water when a peanut was burned. We multiplied the mass of the water (1200 g), the specific heat capacity of water, and the change in temperature (5.25°C) to find the heat absorbed. This type of calculation is the basis of many experiments and calculations in chemistry and physics, especially those involving heat transfer.

Exploring Energy Conversion

Energy conversion is the process of changing one form of energy into another. In our exercise, we first calculated the heat in joules absorbed by the water. Then, we converted this into calories and Calories, which are other units of energy measurement.

- One important conversion factor to remember is: 1 calorie = 4.184 joules. - Another is: 1 Calorie (with an uppercase 'C', often referred to as a kilocalorie) = 1000 calories.

These conversions are useful when dealing with everyday energy issues. For instance, calories and Calories are often used in food energy content and diet planning. The energy conversion in this problem helps us properly understand how the energy from burning the peanut compares in different units, which is crucial for making consistent scientific measurements.

Thermal Energy Calculation Fundamentals

Thermal energy calculation especially involves finding out how much heat is transferred in processes. The formula we used is: \[ Q = mc\Delta T \] where:

• \( Q \) is the heat absorbed or released
• \( m \) is the mass of the substance
• \( c \) is the specific heat capacity
• \( \Delta T \) is the temperature change

In our example, the thermal energy calculations showed us that 26328 joules of energy were released from the peanut. Understanding how to calculate thermal energy is crucial to making sense of how energy is transferred between systems.

Whether you're cooking, designing cooling systems, or studying the climate, mastering these calculations will enable you to better interpret and influence energy changes in various contexts. This skill is particularly relevant in scientific fields concerning heat flow and energy efficiency.