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) \(\left(\delta_{U} / \delta_{\mathrm{T}}\right)_{\mathrm{v}}\) (b) \(\left(\delta_{H} / \delta_{T}\right)_{v}\) (c) \(\left(\delta_{U} / \delta_{\mathrm{T}}\right)_{\mathrm{p}}\) (d) \(\left(\delta_{H} / \delta_{\mathrm{T}}\right)_{\mathrm{p}}\)

Step-by-Step Solution

Answer

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

Step 1: Understanding the Concepts

Molar heat capacities, such as at constant pressure \(C_p\) and at constant volume \(C_v\), represent the amount of heat required to change the temperature of one mole of a substance by one degree at constant pressure and volume, respectively. \(C_p\) and \(C_v\) are defined as the changes in enthalpy and internal energy with respect to temperature at constant pressure and volume, respectively.

Step 2: Identifying Formula for Molar Heat Capacity at Constant Pressure

The molar heat capacity at constant pressure \(C_p\) is defined as the rate of change of enthalpy \(H\) with respect to temperature \(T\) at constant pressure. Mathematically, \(C_p = \left( \frac{\delta H}{\delta T} \right)_p\).

Step 3: Analyzing Answer Choices

We review the provided options: - (a) \(\left( \frac{\delta U}{\delta T} \right)_v\) is related to internal energy change at constant volume, for \(C_v\).- (b) \(\left( \frac{\delta H}{\delta T} \right)_v\) is enthalpy change at constant volume.- (c) \(\left( \frac{\delta U}{\delta T} \right)_p\) involves internal energy at constant pressure.- (d) \(\left( \frac{\delta H}{\delta T} \right)_p\) is indeed enthalpy change at constant pressure, matching the definition of \(C_p\).

Step 4: Confirming the Correct Answer

From Step 2, we know \(C_p\) is defined as \(\left( \frac{\delta H}{\delta T} \right)_p\), which matches option (d). Therefore, option (d) is the correct choice for molar heat capacity at constant pressure.

Key Concepts

Enthalpy

Enthalpy is a fundamental concept in thermodynamics. It represents the total heat content of a system. This is crucial when discussing the heat exchange between a system and its surroundings. Enthalpy, denoted by the symbol \( H \), varies with conditions such as temperature and pressure. It is especially important in processes occurring at constant pressure.

In mathematical terms, enthalpy is expressed as \( H = U + PV \), where \( U \) is the internal energy, \( P \) is the pressure, and \( V \) is the volume. This equation shows that enthalpy combines energy stored within the system and the energy required to expand against atmospheric pressure. In simpler terms, enthalpy can be thought of as the energy the system can release or absorb without doing work on the environment.

Internal Energy

Internal energy is another key concept in understanding thermodynamic systems. This form of energy is the total of all microscopic forms of energy in a substance or system. It is denoted by the symbol \( U \) and includes energies such as kinetic and potential energy at the molecular level.

When a system experiences a change in conditions, such as temperature or volume, its internal energy changes. These changes are often analyzed under constant conditions like constant volume. Internal energy is crucial in explaining why certain processes either absorb or release energy when the system's temperature changes.

• Relates to heat flow: A higher internal energy often means more energy available for heat transfer.
• Influences reaction outcomes: In chemical reactions, changes in internal energy can drive the process.

Temperature Change

Temperature change is a simple but important concept in thermodynamics. It tells us how much the temperature of a system varies as it absorbs or releases heat. This change is usually measured in degrees Celsius or Kelvin.

The molar heat capacity of a substance is closely related to temperature change. It specifies how much energy is required to change the temperature of one mole of a substance by one degree. When a system undergoes a temperature change, the energy exchanges can be calculated using the formula:
\[ q = mc\Delta T \]
where \( q \) is the heat exchange, \( m \) is the mass, \( c \) is the specific heat capacity, and \( \Delta T \) is the temperature change.

• This helps determine the energy needed for heating or cooling systems.
• It also indicates the system's response to added or removed heat.

Constant Pressure

Constant pressure conditions are pivotal when discussing enthalpy and heat capacity. A constant pressure process ensures that even though other variables like temperature or volume might change, the pressure remains unchanged.

In chemistry, especially for reactions occurring in open containers, constant pressure is common since the atmospheric pressure doesn't change dramatically. Under these conditions, the concept of molar heat capacity at constant pressure \( C_p \) comes into play. It is defined by:
\[ C_p = \left( \frac{\delta H}{\delta T} \right)_p \]
This means \( C_p \) represents how enthalpy changes with temperature at constant pressure, making it a vital measure of how much heat is required for a temperature change.

Constant Volume

When studying thermodynamics, the concept of constant volume is crucial. In these processes, the system's volume does not change, meaning any heat added or removed does not perform work in moving a boundary against a constant pressure. This makes analyzing internal energy simpler.

Molar heat capacity at constant volume, denoted as \( C_v \), describes how the internal energy of a system changes with temperature at constant volume. It is expressed as:
\[ C_v = \left( \frac{\delta U}{\delta T} \right)_v \]
In essence, \( C_v \) helps us understand how much heat is needed to raise the temperature without changing the system's volume, providing insights into energy transfer and molecular behavior in the system.