Question

2.25 moles of an ideal gas with \(C_{V, m}=5 R / 2\) are transformed from an initial state \(T=680 . \mathrm{K}\) and \(P=1.15\) bar to a final state \(T=298 \mathrm{K}\) and \(P=4.75\) bar. Calculate \(\Delta U, \Delta H,\) and \(\Delta S\) for this process.

Short answer

For this process with 2.25 moles of an ideal gas, the changes are as follows: \(\Delta U = -1845R\), \(\Delta H = -2583R\), and \(\Delta S \approx -6.97R\).

Step-by-step solution

Calculate Change in Internal Energy (\(\Delta U\))

For an ideal gas, the change in internal energy, \(\Delta U\), depends solely on the difference in temperature between the initial and final states and can be calculated using the formula: \[\Delta U = n C_{V, m}(T_2 - T_1)\] Given the values: - \(n = 2.25 \text{ moles}\) - \(C_{V, m} = \dfrac{5R}{2}\) - \(T_1 = 680 \ \mathrm{K}\) - \(T_2 = 298 \ \mathrm{K}\) Substitute the given values into the formula: \[\Delta U = 2.25 \cdot \dfrac{5R}{2}(298 - 680)\] Calculate the change in internal energy: \[\Delta U = -1845R\]

Calculate Change in Enthalpy (\(\Delta H\))

For an ideal gas, the change in enthalpy, \(\Delta H\), can be calculated using the molar heat capacity at constant pressure (\(C_{p, m}\)) and the difference in temperature between the initial state and the final state. However, we are given the value of \(C_{V, m}\) only. We can use the following relation between \(C_{p, m}\) and \(C_{V, m}\): \[C_{p, m} = C_{V, m} + R\] Substitute the given value of \(C_{V, m}\): \[C_{p, m} = \dfrac{5R}{2} + R\] Now we have \(C_{p, m} = \dfrac{7R}{2}\) and can calculate the change in enthalpy using the formula: \[\Delta H = n C_{p, m}(T_2 - T_1)\] Substitute the values: \[\Delta H = 2.25 \cdot \dfrac{7R}{2}(298 - 680)\] Calculate the change in enthalpy: \[\Delta H = -2583R\]

Calculate Change in Entropy (\(\Delta S\))

For an ideal gas, the change in entropy, \(\Delta S\), can be calculated using the formula: \[\Delta S = n C_{V, m} \ln{\dfrac{T_2}{T_1}} + n R \ln{\dfrac{P_2}{P_1}}\] Given the values: - \(P_1 = 1.15 \ \mathrm{bar}\) - \(P_2 = 4.75 \ \mathrm{bar}\) Substitute the values into the formula: \[\Delta S = 2.25 \cdot \dfrac{5R}{2} \ln{\dfrac{298}{680}} + 2.25R \ln{\dfrac{4.75}{1.15}}\] Calculate the change in entropy: \[\Delta S \approx -6.97R\] Now we have the values for \(\Delta U\), \(\Delta H\), and \(\Delta S\) for this process: - \(\Delta U = -1845R\) - \(\Delta H = -2583R\) - \(\Delta S \approx -6.97R\)

Key concepts

Change in Internal Energy

When dealing with ideal gases, the change in internal energy (\textbf{\(\text{\Delta U}\)}) is an essential concept in understanding how energy is transferred in the form of heat or work during gas processes. The internal energy is a property related only to the temperature of an ideal gas, independent of volume or pressure changes. This is because ideal gases are assumed to have no intermolecular forces, and the particles are in perpetual motion, with their kinetic energy correlating with temperature. To calculate the change in internal energy, use the equation:

\[\Delta U = n C_{V, m}(T_2 - T_1)\]
Here, \(n\) is the number of moles, \(C_{V, m}\) is the molar heat capacity at constant volume, \(T_1\) is the initial temperature, and \(T_2\) is the final temperature. The molar heat capacity at constant volume represents the amount of energy required to raise the temperature of one mole of the gas by one degree while keeping the volume constant. Since the problem provides \(C_{V, m}\) and the temperatures, you can calculate the energy change directly, as seen in the exercise solution.

Change in Enthalpy

The enthalpy change (\textbf{\(\text{\Delta H}\)}) in ideal gas processes reflects energy changes related to heat under constant pressure. It's interconnected with the concepts of internal energy and work done by the system. To determine the enthalpy change, one would typically use the molar heat capacity at constant pressure (\textbf{\(C_{p, m}\)}), which relates to the energy needed to raise the temperature of one mole of gas by one degree at constant pressure. The relationship between the molar heat capacities is given by the equation:

\[C_{p, m} = C_{V, m} + R\]
Adding the universal gas constant \(R\) to the molar heat capacity at constant volume transitions the calculation from an isochoric to an isobaric perspective. Once \(C_{p, m}\) is known, the change in enthalpy is calculated by:

\[\Delta H = n C_{p, m}(T_2 - T_1)\]
As can be seen in the provided solution, the change in enthalpy for an ideal gas only depends on the temperature change and the molar heat capacity at constant pressure.

Change in Entropy

Entropy (\textbf{\(\text{\Delta S}\)}) is a measure of disorder or randomness in a system and is a fundamental concept in thermodynamics. The entropy of a system increases when energy is distributed into it in a reversible process. For an ideal gas, the change in entropy is affected by both temperature and pressure changes. The formula to calculate the change in entropy of an ideal gas is:

\[\Delta S = n C_{V, m} \text{ln} \left(\frac{T_2}{T_1}\right) + n R \text{ln} \left(\frac{P_2}{P_1}\right)\]
The first term considers the temperature change at constant volume, and the second term accounts for the pressure change at constant temperature. In essence, entropy change reflects the energy redistribution within the system when it moves from one state to another. The calculations performed in the step-by-step solution showcase the direct application of this formula, providing insights into the thermodynamic behavior of the gas during the transition between the stated conditions.

Molar Heat Capacity

Molar heat capacity is an intrinsic property of a substance that defines how much heat energy is needed to raise the temperature of one mole of a substance by one degree Celsius. There are two types to consider: the molar heat capacity at constant volume (\(C_{V, m}\)) and at constant pressure (\(C_{p, m}\)).

For an ideal gas, the molar heat capacity at constant volume represents the amount of heat required to raise the temperature while maintaining a fixed volume. It is related to translational and, depending on the gas, rotational and vibrational motions of the molecules. The ability to distinguish between \(C_{V, m}\) and \(C_{p, m}\) is vital in thermodynamics because it affects how the energy of the gas is partitioned between heat and work in different processes. Notably, these capacities are related to universal gas constant \(R\) through the equation \(C_{p, m} = C_{V, m} + R\), showing that holding pressure constant requires more energy input than volume because work is done against the external pressure.