Question

Asuming that water vapour is an ideal gas, the internal energy change \((\Delta U)\) when 1 mol of water is vapourized at 1 bar pressure and \(100^{\circ} \mathrm{C}\), (Given: Molar enthalpy of vaporization of water at 1 bar and \(373 \mathrm{~K}=41 \mathrm{~kJ} \mathrm{~mol}^{-1}\) and \(\left.\mathrm{R}=8.3 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\right)\) will be: (a) \(3.7904 \mathrm{~kJ} \mathrm{~mol}^{-1}\) (b) \(37.904 \mathrm{~kJ} \mathrm{~mol}^{-1}\) (c) \(41.00 \mathrm{~kJ} \mathrm{~mol}^{-1}\) (d) \(4.100 \mathrm{~kJ} \mathrm{~mol}^{-1}\)

Short answer

The internal energy change is \(37.904 \, \text{kJ/mol}\) (Option b).

Step-by-step solution

Understand the given data

We have the molar enthalpy of vaporization \(\Delta H = 41 \, \text{kJ} \, \text{mol}^{-1}\), and we are given the gas constant \(R = 8.3 \, \text{J} \, \text{mol}^{-1} \, \text{K}^{-1}\). Temperature is \(100^\circ \text{C} = 373 \text{ K}\) and pressure is 1 bar.

Apply the thermodynamic relation between enthalpy and internal energy

The relation between the change in enthalpy \(\Delta H\) and the change in internal energy \(\Delta U\) for a process at constant pressure is given by the equation: \(\Delta H = \Delta U + P \Delta V\).

Calculate \(P \Delta V\) using the ideal gas equation

Since water vapor is assumed to be an ideal gas, we apply the ideal gas law to find \(P \Delta V\): \(PV = nRT\). For 1 mole, \(P \Delta V = R \Delta T\). Given \(R = 8.3 \, \text{J} \, \text{mol}^{-1} \, \text{K}^{-1}\) and \(T = 373 \, \text{K}\), \(P \Delta V = 8.3 \, \text{J/mol} \times 373 \, \text{K} = 3085.9 \, \text{J/mol} = 3.086 \, \text{kJ/mol}\).

Solve for \(\Delta U\)

Rearrange the enthalpy and internal energy equation to solve for \(\Delta U\): \(\Delta U = \Delta H - P \Delta V\). Substituting the given values: \(\Delta U = 41.0 \, \text{kJ/mol} - 3.086 \, \text{kJ/mol}\).

Calculate \(\Delta U\)

Perform the subtraction: \(\Delta U = 41.0 - 3.086 = 37.914 \, \text{kJ/mol}\). Round to match given options.

Match the result with options

The result \(\Delta U = 37.914 \, \text{kJ/mol}\) rounds to \(37.904 \, \text{kJ/mol}\), which corresponds to option (b).

Key concepts

Ideal Gas Law

The Ideal Gas Law is an essential concept in thermodynamics that helps us understand the behavior of gases under various conditions. This law is represented by the equation \(PV = nRT\), where:

• \(P\) is the pressure of the gas
• \(V\) is the volume
• \(n\) is the number of moles
• \(R\) is the ideal gas constant, approximately 8.314 J/(mol·K)
• \(T\) is the temperature in Kelvin

To apply the Ideal Gas Law, it's crucial that the gas behaves ideally, meaning it experiences no interactions between molecules other than elastic collisions. In our context, the problem assumes water vapor follows this law, enabling us to calculate changes in volume as temperature changes, using a constant pressure. By substituting known values for \(R\), \(T\), and the quantity of substance in moles, you can determine factors like \(P \Delta V\) for processes like vaporization.

Internal Energy

Internal energy is a fundamental thermodynamic property representing the total energy contained within a system. It encompasses both the kinetic energy, which includes the movement of individual molecules, and potential energy, associated with molecular forces. The change in internal energy, \(\Delta U\), is an important part of the energy balance for any process. According to the first law of thermodynamics, energy can neither be created nor destroyed, only transferred or converted from one form to another. For an ideal gas, the change in internal energy can be directly related to the change in temperature, given utilization of a simplified model where only kinetic energy changes since potential interactions are negligible. In calculations, \(\Delta U\) can be related to enthalpy and expansion work by the formula: \(\Delta U = \Delta H - P \Delta V\). This relationship helps determine the energy change associated with phase transitions, such as vaporization.

Enthalpy Change

Enthalpy is another crucial thermodynamic quantity that describes heat content or potential energy contained within a system. It's denoted by \(H\) and combines the system's internal energy \(U\) with the energy related to its pressure and volume, through the relationship \(H = U + PV\). When a system undergoes a process at constant pressure, like vaporizing water, the change in enthalpy \(\Delta H\) represents significant heat exchanged. The molar enthalpy of vaporization refers specifically to the heat required to transform one mole of a substance from liquid to gas at a constant pressure. In the context of this problem, the value provided for \(\Delta H\) aids in calculating internal energy change. This calculation is executed using the formula: \(\Delta U = \Delta H - P\Delta V\), where \(P\Delta V\) accounts for the work done during the expansion of the gas. By understanding these principles, students can correlate the heat exchange with changes in state and energy forms.