Question

The molar heat capacity of water at constant pressure, \(\mathrm{C}\), is \(75 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\). When \(1.0 \mathrm{~kJ}\) of heat is supplied to \(100 \mathrm{~g}\) of water which is free to expand, the increase in temperature of water is (a) \(4.8 \mathrm{~K}\) (b) \(6.6 \mathrm{~K}\) (c) \(1.2 \mathrm{~K}\) (d) \(2.4 \mathrm{~K}\)

Short answer

The increase in temperature of water is 2.4 K, corresponding to option (d).

Step-by-step solution

Identify Given Data

The molar heat capacity of water at constant pressure, \(C_p \), is given as 75 J/K mol. Heat supplied, \(q\), is 1.0 kJ, which is 1000 J. The mass of water is 100 g.

Convert Mass to Moles

To find the number of moles, we use the formula: \(\text{moles} = \frac{\text{mass}}{\text{molar mass}} \). The molar mass of water is approximately 18 g/mol. So, \(\text{moles of water} = \frac{100 \text{ g}}{18 \text{ g/mol}} = 5.56 \text{ moles} \).

Use Heat Capacity Formula

The formula for heat is \(q = nC_p \Delta T\), where \(n\) is the number of moles, \(C_p\) is the molar heat capacity, and \(\Delta T\) is the temperature change. Rearranging to solve for \(\Delta T\), we get \(\Delta T = \frac{q}{nC_p}\).

Plug in the Values

Substitute the values into the equation: \(\Delta T = \frac{1000 \text{ J}}{5.56 \text{ moles} \times 75 \text{ J/K mol}} = 2.4 \text{ K}\).

Select the Correct Option

The calculated increase in temperature is 2.4 K. This corresponds to option (d).

Key concepts

Heat Transfer

Heat transfer is a fundamental concept in thermodynamics, describing how heat energy moves from one body or system to another. When a substance like water is supplied with heat, this energy causes its molecules to move more rapidly, which is often sensed as an increase in temperature. In the context of our original problem, 1.0 kJ of energy, or 1000 J, is introduced to 100 g of water. The goal is to determine how this heat affects the water's temperature. This process involves assessing the energy exchange per mole of the substance, which is described by the molar heat capacity. Understanding heat transfer is crucial for many practical applications ranging from cooking to industrial processes. It affects how efficiently systems can be designed to maintain or change temperature. Heat transfer can occur in three ways: conduction, convection, and radiation, each with its unique characteristics:

• Conduction: Direct heat flow through a medium, requiring a temperature gradient.
• Convection: Fluid motion helps transfer heat, seen in gases and liquids.
• Radiation: Heat transfer via electromagnetic waves, not requiring a medium.

Each method depends on different conditions and materials involved, making a deep understanding of these modes essential in thermodynamics.

Temperature Change

The temperature change in a substance, such as water, results from its ability to absorb and store thermal energy effectively. This change, \(\Delta T\), can be influenced by several factors, including the substance's specific heat capacity, the amount of heat supplied, and the mass of the substance.In our exercise, heat is supplied to water which then leads to an increase in temperature. We began with a known quantity of heat and calculated a temperature increase using the formula for heat capacity:\[ \Delta T = \frac{q}{nC_p} \]Where \(q\) is the heat added, \(n\) is the number of moles, and \(C_p\) is the molar heat capacity. This calculation involves determining the substance’s molar heat capacity and accurately converting the units of heat energy and mass to determine the number of moles. Molar heat capacity (C_p) reflects how much heat is needed to raise the temperature of a mole of the substance by one degree Kelvin. For water, a common value is \(75 \, \text{J/K mol}\), helping to predict the temperature change when a specific amount of heat is introduced. These principles help in designing systems for heating or cooling effectively and understanding natural processes like climate dynamics.

Thermodynamics Concepts

Thermodynamics deals with the principles governing heat, work, and energy within systems. These principles are universal and can be applied to a variety of scientific and engineering problems.The original exercise showcases fundamental ideas such as heat capacity and energy transfer, which are part of the four laws of thermodynamics. The first law, related directly to our task, states that energy cannot be created or destroyed, only converted from one form to another. In simple terms:\[ \Delta U = q - W \]Here, \(\Delta U\) is the change in internal energy, \(q\) is heat added to the system, and \(W\) is the work done by the system. Applied to our problem, this principle helps to equate the supplied heat energy with the resulting change in temperature of the water.Understanding these thermodynamic concepts allows us to predict how a substance behaves when it interacts with heat, thus supporting a broad spectrum of applications, from industrial design to environmental science. These foundational ideas are central to fields like mechanical engineering, chemical processing, and environmental management, and mastering them is crucial for advancing in sciences.