Problem 5 - A Complete Resource Book in Chemistry for JEE Main

Question

Molar heat capacity at constant \(\mathrm{P}\) for a substance is equal to (a) \(\left(\delta_{\mathrm{U}} / \delta_{\mathrm{T}}\right)_{\mathrm{v}}\) (b) \(\left(\delta_{H} / \delta_{\mathrm{T}}\right)_{\mathrm{V}}\) (c) \(\left(\delta_{\mathrm{u}} / \delta_{\mathrm{T}}\right)_{\mathrm{p}}\) (d) \(\left(\delta_{\mathrm{H}} / \delta_{\mathrm{T}}\right)_{\mathrm{P}}\)

Step-by-Step Solution

Answer

The correct answer is (d) \( \left( \frac{\delta H}{\delta T} \right)_P \).

Step 1: Understanding the Concept

The molar heat capacity at constant pressure, denoted as \( C_p \), is defined as the amount of heat required to raise the temperature of one mole of a substance by one degree Celsius at constant pressure. Mathematically, it is represented as \( C_p = \left( \frac{\delta H}{\delta T} \right)_P \), where \( H \) is enthalpy and \( T \) is temperature.

Step 2: Analyzing the Options

We need to determine which option correctly represents the expression for molar heat capacity at constant pressure, \( C_p \). Each option presents a derivative involving a thermodynamic quantity with respect to temperature, either at constant volume (\( V \)) or constant pressure (\( P \)).

Step 3: Evaluating Each Option

We evaluate the given options: - Option (a) represents \( \left( \frac{\delta U}{\delta T} \right)_v \), which is the heat capacity at constant volume, \( C_v \), not \( C_p \).- Option (b) \( \left( \frac{\delta H}{\delta T} \right)_V \) involves enthalpy but is defined at constant volume, so it is not \( C_p \).- Option (c) \( \left( \frac{\delta u}{\delta T} \right)_p \) involves internal energy \( u \) at constant pressure. This is not relevant to \( C_p \).- Option (d) \( \left( \frac{\delta H}{\delta T} \right)_P \) is the correct definition for \( C_p \), involving enthalpy at constant pressure.

Step 4: Selecting the Correct Answer

Given the analysis, option (d) \( \left( \frac{\delta H}{\delta T} \right)_P \) correctly represents the molar heat capacity at constant pressure.

Key Concepts

Molar Heat Capacity

Understanding molar heat capacity is essential in thermodynamics to grasp how substances interact with heat. Molar heat capacity, especially at constant pressure, is a key parameter that helps in determining how much energy (in the form of heat) is required to increase the temperature of one mole of a substance by one degree Celsius. This is symbolized as \(C_p\). The expression for molar heat capacity at constant pressure is given by \(C_p = \left(\frac{\delta H}{\delta T} \right)_P\), where \(H\) signifies enthalpy and \(T\) represents temperature.
The distinction between constant pressure and constant volume is crucial, as they affect how energy is stored in a system. At constant pressure, energy can do work by expanding and creating volume change. This work, along with a change in temperature, contributes to the definition of molar heat capacity in this context.

Enthalpy

Enthalpy is a thermodynamic quantity expressing the total heat content of a system. It's particularly significant because it considers both internal energy and the energy required to make room for it by displacing its environment. This is why \(H\), or enthalpy, is so closely linked with processes occurring at constant pressure.
Mathematically, enthalpy can be represented as \(H = U + PV\), where \(U\) is the internal energy and \(PV\) represents the pressure-volume work done by the system. It’s crucial for understanding how energy transfers in chemical reactions, especially those occurring open to the atmosphere. This concept is integral in calculating heat changes at constant pressure, reinforcing why \(\left(\frac{\delta H}{\delta T} \right)_P\) accurately describes molar heat capacity.

Constant Pressure

The idea of constant pressure in thermodynamics has applications across various scenarios, from atmospheric processes to chemical reactions. At constant pressure, the system is allowed to expand and perform work, as the environment pressure remains unchanged. This setting is critical when dealing with gas expansions or compressions since it directly influences the enthalpy and heat capacity calculations.
Processes that occur under constant pressure are common because many laboratory and real-world reactions happen at atmospheric pressure. This makes calculations involving constant pressure more realistic and pertinent for practical applications, connecting directly with concepts like enthalpy and molar heat capacity.

Internal Energy

Internal energy, denoted as \(U\), is a core concept in thermodynamics that represents the total of all energies within a system. This includes kinetic energy from atomic movements and potential energy from atomic interactions. For a closed system, changes in internal energy can include heat added or removed, and work done on or by the system.
At constant volume, internal energy changes can be simply defined mathematically. However, under constant pressure, it informs part of the broader concept of enthalpy. The relation \(H = U + PV\) connects how internal energy transitions when a system not only absorbs heat, but does work on its surroundings. This makes understanding internal energy pivotal in comprehending how systems behave under different thermodynamic conditions.