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

No option matches \(\Delta E = 1000 \text{ J}\), indicating a potential error in the problem statement or its answer options.

Step 1: Calculating the number of moles

First, determine the number of moles of oxygen. The molecular weight of oxygen (O₂) is 32 g/mol. Given 64 g of oxygen, calculate the number of moles using the formula: \[ n = \frac{\text{mass}}{\text{molar mass}} = \frac{64 \, \text{g}}{32 \, \text{g/mol}} = 2 \, \text{mol} \]

Step 2: Understanding the concept of constant volume process

In a constant volume process, the change in internal energy \(\Delta E\) is given by the formula: \[ \Delta E = n C_v \Delta T \] 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: Calculating the change in temperature

Calculate the change in temperature \(\Delta T\) from the initial and final temperatures given: \[ \Delta T = 100^{\circ} \text{C} - 0^{\circ} \text{C} = 100 \, \text{C} = 100 \, \text{K} \] (The change in Celsius is equal to the change in Kelvin).

Step 4: Calculating the change in internal energy

Plug in the values into the formula for \(\Delta E\): \[ \Delta E = n C_v \Delta T = 2 \, \text{mol} \times 5 \, \text{J/K/mol} \times 100 \, \text{K} = 1000 \, \text{J} \] Upon re-evaluation this figure seems sufficient given prior calculations, hence check against multiple choice options.

Key Concepts

Constant Volume Process

In thermodynamics, a constant volume process refers to a situation where the volume of a system does not change over time. This means no work is done by the system since work in thermodynamics is defined as pressure times the change in volume. No volume change results in zero work, simplifying the calculations of energy changes.
During a constant volume process, any heat added to or removed from the system will simply result in a change in the system's internal energy, without affecting the volume. The formula to calculate the change in internal energy (DE) during a constant volume process is:

• DE = n CV DT where n is the number of moles of the substance, CV is the molar heat capacity at constant volume, and DT is the change in temperature.

This specific scenario maintains simplicity because only heat transfer affects internal energy, without complications from volume changes.

Heat Capacity

Heat capacity is a fundamental concept in thermodynamics that quantifies the amount of heat required to change a system's temperature by a certain amount. Different types of heat capacity exist, and in the context of a constant volume process, we are particularly interested in the molar heat capacity at constant volume, denoted as CV.

Heat capacity at constant volume measures how much heat is needed to increase the temperature of one mole of a substance by one Kelvin without any change in volume. It is given in units of Joules per Kelvin per mole (J/K/mol). For example, in this problem, the oxygen's heat capacity at constant volume is provided as 5 J/K/mol.

• This allows us to calculate the amount of internal energy change based on how much the temperature changes.
• The constant is unique for each gas, based on its molecular structure and interactions.

Change in Internal Energy

The change in internal energy (DE) of a system is a vital aspect of studying thermodynamics, especially during a constant volume process. It represents the total increase or decrease in the system's energy, considering all the modes of energy storage, like translational, rotational, and in some cases, vibrational energy of molecules.

• In a constant volume process, this change is determined solely by the heat added or removed, given no work is done.
• The calculation involves multiplying the number of moles, the molar heat capacity at constant volume (CV), and the change in temperature (DT).

Using the formula DE = n CV DT, we calculate the change as 1000 J for the given set up of 2 moles of oxygen, a heat capacity of 5 J/K/mol, and a temperature change of 100 K.
The change is significant as it assists in understanding how energy interacts with matter under conditions of fixed volume, highlighting the intrinsic thermodynamic property that guides these energy interactions.