The molar heat capacity of water at constant pressure, \(C_{P}\) is \(75 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\). When \(10 \mathrm{~kJ}\) of heat is supplied to \(1 \mathrm{~kg}\) water which is free to expand, the increase in temperature of water is (a) \(2.4 \mathrm{~K}\) (b) \(4.8 \mathrm{~K}\) (c) \(3.2 \mathrm{~K}\) (d) \(10 \mathrm{~K}\)

Thermodynamics is a branch of physics that deals with the relationships between heat and other forms of energy. At its core, it involves studying how energy is converted from one form to another and how it affects matter. It is governed by several fundamental laws, which include the first law, often referred to as the law of energy conservation. This law states that energy cannot be created or destroyed, only transferred or converted from one form to another.

The exercise in question touches upon the first law of thermodynamics by examining how the transfer of heat (\( q \)), to water results in an increase in temperature. It is essential to understand that the thermal energy transferred to the water will cause the water molecules to move more rapidly, leading to a rise in temperature. This temperature change (\( \triangle T \)) is an important concept in thermodynamics as it is a measurable physical quantity that can often be directly related to the quantity of energy transferred.

Molar heat capacity (\( C_P \)) is a significant concept within thermodynamics and is a measure of the amount of heat required to raise the temperature of one mole of a substance by one Kelvin at constant pressure. It is intrinsic to understanding how different materials respond to heat. Molar heat capacity is a specific property of a material and is critical for predicting temperature changes when energy is added or removed.

In the textbook exercise, the molar heat capacity of water is given, allowing for calculations involving heat transfer and temperature changes. Knowing the molar heat capacity is necessary for targeting the precise energy requirements for temperature adjustments in various applications, from industrial processes to everyday cooking. It's why we can predict that adding a certain amount of heat to a specific mass of water will result in a definable increase in temperature.

Heat transfer refers to the process of thermal energy moving from a hotter object to a cooler one. It is a fundamental concept within thermodynamics and occurs in various ways, including conduction, convection, and radiation. In our daily lives, we witness heat transfer when a warm coffee cup cools down to room temperature or when the sun heats the surface of the Earth.

In the context of the exercised problem, heat transfer is explored quantitatively by applying a specific amount of heat to water and observing the resultant change in temperature. The heat transfer formula \( q = nC_P\triangle T \) connects the dots between the heat supplied (\( q \)), the substance's molar heat capacity (\( C_P \)), and the resultant temperature change (\( \triangleT \)). By manipulating the formula, we can solve for the unknown temperature change once the amount of heat added and the molar heat of the substance are known. This process not only illustrates how temperature change is directly related to heat transfer but also how substances react uniformly based on their individual molar heat capacities.