Problem 81 - A Complete Resource Book in Chemistry for JEE Main

Question

What is the value of \(\Delta \mathrm{E}\), when \(64 \mathrm{~g}\) oxygen is heated from \(0^{\circ} \mathrm{C}\) to \(100^{\circ} \mathrm{C}\) at constant volume? \(\left(\mathrm{C}_{\mathrm{v}}\right.\) on an average is \(5 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\) ) (a) \(1500 \mathrm{~J}\) (b) \(1800 \mathrm{~J}\) (c) \(2000 \mathrm{~J}\) (d) \(2200 \mathrm{~J}\)

Step-by-Step Solution

Answer

The calculated \( \Delta E \) is 1000 J, but considering practical context, re-evaluate for possibility of other calculations (like \( \Delta H \)) if options mismatch.

Step 1: Determine the number of moles

Oxygen (O_2) has a molar mass of approximately 32 g/mol. Given that we have 64 g of oxygen, we can determine the number of moles using the formula \( \text{moles} = \frac{{\text{mass}}}{{\text{molar mass}}} \).\[\text{moles} = \frac{64 \text{ g}}{32 \text{ g/mol}} = 2 \text{ moles}\]

Step 2: Use the formula for change in internal energy

The formula for the change in internal energy \(\Delta \text{E}\) at constant volume is given by \( \Delta \text{E} = n \cdot C_v \cdot \Delta T \), where \( n \) is the number of moles, \( C_v \) is the molar heat capacity at constant volume, and \( \Delta T \) is the change in temperature.

Step 3: Calculate the change in temperature \(\Delta T\)

\( \Delta T \) is the difference between the final temperature and the initial temperature.\[\Delta T = 100^{\circ} \text{C} - 0^{\circ} \text{C} = 100 \text{ C}\]Note that Celsius and Kelvin scales have the same magnitude for temperature differences.

Step 4: Calculate \(\Delta \text{E}\) using the formula

Substitute the values \( n = 2 \text{ moles} \), \( C_v = 5 \text{ J/mol.K} \), and \( \Delta T = 100 \text{ K} \) into the change in internal energy formula.\[\Delta \text{E} = 2 \times 5 \times 100 = 1000 \text{ J}\]

Step 5: Check the options

None of the given options match the calculated \( \Delta \text{E} = 1000 \text{ J} \). However, the problem must involve the calculation of \( \Delta H \) instead of \( \Delta E \), as indicated by the provided options.Recalculate considering this new understanding, or check if any data is missing that aligns to the known options.

Key Concepts

Internal Energy

The internal energy of a system refers to the total energy contained within it. It's intrinsic to the system and includes potential and kinetic energies at the microscopic level. In the context of thermodynamics, internal energy is a crucial concept that is often denoted by the symbol \( E \) or \( U \). When you heat a substance, such as oxygen at a constant volume, its internal energy changes. This change in internal energy \( \Delta E \) is influenced by the heat capacity of the substance and the temperature change it undergoes.

In our example, the change in internal energy was calculated using the formula:

• \( \Delta E = n \cdot C_v \cdot \Delta T \)

where:

• \( n \) is the number of moles,
• \( C_v \) is the molar heat capacity at constant volume,
• \( \Delta T \) is the change in temperature.

Understanding how internal energy behaves under different conditions helps predict how changes in temperature or pressure affect a system's state, making it a fundamental concept in thermochemistry.

Molar Heat Capacity

Molar heat capacity is a property of materials that indicates how much heat is required to raise the temperature of one mole of the substance by one degree Kelvin (or Celsius, given the equivalence in temperature difference). When considering constant volume processes, the molar heat capacity is often represented as \( C_v \).

In our exercise, the molar heat capacity of oxygen is given as \( 5 \text{ J/mol.K} \). This figure tells us how much energy is needed to heat one mole of oxygen by one degree, provided the volume does not change. This value is crucial for calculating the change in internal energy when you know the temperature increase and the number of moles involved.

The formula that incorporates molar heat capacity to find the change in internal energy is:

• \( \Delta E = n \cdot C_v \cdot \Delta T \)

This calculation helps chemists and physicists understand and predict how different substances behave when subject to temperature changes. It also reinforces the importance of specific heat capacities in energy transfer processes.

Thermochemistry

Thermochemistry studies the heat involved in chemical processes and physical changes of state. It’s a branch of thermodynamics that focuses on the measurement and interpretation of heats of reaction and change, crucial for understanding energy transfer during chemical reactions and phase transitions.

In simple terms, it helps predict whether a reaction will absorb or release energy, influencing how substances interact with heat. Thermochemistry uses concepts like enthalpy, entropy, and internal energy to explain these interactions.

For exercises involving heating, like the one presented, thermochemistry principles help determine changes in internal energy or enthalpy. These calculations are essential for:

• Designing energy-efficient systems,
• Creating thermodynamic models,
• Understanding the energy balance in chemical reactions,
• Predicting reaction outcomes based on heat exchange.

Thermochemistry connects the macroscopic observations of heat and work to microscopic properties, offering a comprehensive view of energy flow in chemical and physical changes.