What's the specific heat of a material if it takes \(7.5 \mathrm{kJ}\) to increase the temperature of a \(1-\mathrm{kg}\) sample by \(3.0^{\circ} \mathrm{C} ?\)

When we talk about thermal energy transfer, we are referring to the movement of heat from one body or substance to another. This can occur in three fundamental ways: conduction, convection, and radiation. In our example, when an object with mass is heated, thermal energy is transferred into it, increasing the internal energy of the material, hence raising its temperature.

Using the exercise as a guide, we can see that the given material absorbs 7.5 kJ of heat energy. This means that thermal energy has been transferred to the material, causing an increase in temperature. The process of thermal energy transfer is crucial in determining how much heat is required to change the temperature of a material by a specific amount, which is precisely what the specific heat calculation can tell us.

The concept of temperature change, denoted as \(\Delta T\), is the difference between the final and initial temperatures of a substance. It is generally measured in degrees Celsius (\(°C\)) or Kelvin (\(K\)). In our exercise, the temperature change of the material is 3.0°C.

Understanding the relationship between heat transfer and temperature change is essential for many scientific calculations. In the context of the exercise, the 7.5 kJ of energy transferred to the \(1-kg\) material is solely responsible for its temperature change of 3.0°C. The specific heat, therefore, quantifies the amount of heat needed to raise the temperature of one kilogram of the material by one degree Celsius.

Lastly, heat capacity is a property that tells us how much heat energy a substance can store. In more scientific terms, it's defined as the amount of heat required to raise the temperature of an object by 1°C. However, as we delve a bit deeper, it's crucial to distinguish between heat capacity, which depends on the object’s mass, and specific heat capacity, which is an intrinsic property of a material irrespective of its mass.

In the provided exercise, we're looking for the specific heat capacity (\(c\)) of a material, which is the heat capacity per unit mass. The problem statement leads us to the calculation of the specific heat using the formula \(c = \frac{q}{m\Delta T}\), where \(q\) is the thermal energy transfer in joules (\(J\)), \(m\) is the mass of the substance in kilograms (\(kg\)), and \(\Delta T\) is the temperature change in degrees Celsius (\(°C\)). By dividing the total heat energy by the product of mass and temperature change, we obtain the specific heat capacity, which in this case is \(2500 \, \mathrm{J/kg \cdot C}\).