Question

A 2.25 mole sample of carbon dioxide, for which \(C_{P, m}=37.1 \mathrm{JK}^{-1} \mathrm{mol}^{-1}\) at \(298 \mathrm{K},\) is expanded reversibly and adiabatically from a volume of \(4.50 \mathrm{L}\) and a temperature of \(298 \mathrm{K}\) to a final volume of \(32.5 \mathrm{L}\). Calculate the final temperature, \(q, w, \Delta H,\) and \(\Delta U .\) Assume that \(C_{P, m}\) is constant over the temperature interval.

Short answer

The final temperature of the CO₂ sample after the adiabatic expansion is 198.96 K. There is no heat exchange (q = 0 J), and the work done during the process is 6409.75 J. The internal energy change (ΔU) is also 6409.75 J, and the enthalpy change (ΔH) is -8292.44 J.

Step-by-step solution

Determine the final temperature using the adiabatic process equation

Since the process is adiabatic and reversible, we will use the adiabatic process equation: \((V_i)^{C_P/C_V-1}T_i=(V_f)^{C_P/C_V-1}T_f\) Given, molar heat capacity at constant pressure, \(C_{P, m} = 37.1 J K^{-1} mol^{-1}\), we need molar heat capacity at constant volume, \(C_{V, m}\), to find the final temperature, Tf. As we know, for any ideal gas, \(C_{P, m} = C_{V, m} + R\), where R is the gas constant (8.314 J K^{-1} mol^{-1}). Therefore, \(C_{V, m} = C_{P, m} - R = 37.1 - 8.314 = 28.786 J K^{-1} mol^{-1}\). Now, let's find the factor, \(C_{P, m}/C_{V, m}\) - 1: \(C_{P, m}/C_{V, m} - 1 = \frac{37.1}{28.786} - 1 = 1.2889\) Using the adiabatic process equation, we have: \((V_i)^{1.2889}T_i=(V_f)^{1.2889}T_f\) \(T_f = \frac{(V_i)^{1.2889}T_i}{(V_f)^{1.2889}} = \frac{(4.5)^{1.2889} * 298}{(32.5)^{1.2889}} = 198.96 K\)

Calculate work (w) during an adiabatic expansion

In an adiabatic expansion, work (w) can be calculated by the following formula: \(w = nC_{V, m}(T_i - T_f)\) Substitute the values, we get: \(w = 2.25 * 28.786 * (298 - 198.96) = 2.25 * 28.786 * 99.04 = 6409.75 J\)

Calculate internal energy change (ΔU) and enthalpy change (ΔH)

Since the expansion is adiabatic, there will be no heat exchange, i.e., q = 0. As ΔU = q + w, we can find the internal energy change: \(\Delta U = 0 + 6409.75 = 6409.75 J\) For an ideal gas, enthalpy change (ΔH) can be calculated using the formula: \(\Delta H = nC_{P, m} \Delta T = nC_{P, m} (T_f - T_i)\) Substitute the values: \(\Delta H = 2.25 * 37.1 * (198.96 - 298) = -2.25 * 37.1 * 99.04 = -8292.44 J\) Now, we have all the values required: Final temperature, Tf = 198.96 K Heat, q = 0 J Work, w = 6409.75 J Internal energy change, ΔU = 6409.75 J Enthalpy change, ΔH = -8292.44 J

Key concepts

Ideal Gas

An ideal gas is a theoretical concept that helps simplify the study of gas behaviors. It assumes that gas particles are point masses that do not interact with each other, thereby allowing calculations with straightforward equations. Ideal gases follow the ideal gas law, given by the equation \( PV = nRT \), where

• \(P\) is the pressure
• \(V\) is the volume
• \(n\) is the number of moles
• \(R\) is the gas constant (8.314 J K\(^{-1}\) mol\(^{-1}\))
• \(T\) is the temperature in Kelvin

In an adiabatic process, like the one described in the problem, an ideal gas does not exchange heat with its surroundings. As a result, any change in internal energy is solely due to work done by or on the gas. Calculating the changes in state for ideal gases relies heavily on their specific heat capacities and equations of state.

Molar Heat Capacity

Molar heat capacity is a property of a substance that describes how much heat energy is needed to raise the temperature of one mole of that substance by one Kelvin. For gases, there are two main types:

• **Molar heat capacity at constant volume (\(C_{V,m}\))**: The amount of heat required to increase the temperature of one mole of gas by one Kelvin at constant volume.
• **Molar heat capacity at constant pressure (\(C_{P,m}\))**: The amount of heat required to increase the temperature of one mole of gas by one Kelvin at constant pressure.

In ideal gases, these two capacities are related through the equation \( C_{P,m} = C_{V,m} + R \). This relation arises because work is done by the gas when it expands at constant pressure. In the problem, it's given that the molar heat capacity at constant pressure is 37.1 J K\(^{-1}\) mol\(^{-1}\), allowing us to calculate the molar heat capacity at constant volume as 28.786 J K\(^{-1}\) mol\(^{-1}\). This distinction is crucial for determining the gas's behavior during changes in the process.

Internal Energy Change

The internal energy change (\(\Delta U\)) of a system signifies the net change in the system's internal energy due to work done and heat transferred. In an adiabatic process, where no heat is exchanged (\(q = 0\)), any change in the system's internal energy is exclusively due to the work done on or by the system. This is represented by the first law of thermodynamics: \[ \Delta U = q + w \] This simplifies to \(\Delta U = w\) for adiabatic processes, meaning the change in internal energy equals the work done.In the given exercise, the process involved adiabatic expansion, so the work done was 6409.75 J. Consequently, the internal energy change \(\Delta U\) was also 6409.75 J, indicating that all the energy change was due to work performed without heat exchange.

Enthalpy Change

Enthalpy change (\(\Delta H\)) in a thermodynamic system is a measure of the total heat content change. For an ideal gas undergoing a process, this depends on changes in temperature, specifically calculated when pressure is constant using the equation: \[ \Delta H = nC_{P,m} \Delta T \] Here, \(\Delta T\) is the difference between the final and initial temperatures In the exercise, the initial and final temperatures were 298 K and 198.96 K, respectively. Since the process was adiabatic, the enthalpy change reflects the thermal energy associated with the temperature drop. So, substituting the values gives \(\Delta H = 2.25 \times 37.1 \times (198.96 - 298) = -8292.44 \) J. The negative sign indicates a decrease in enthalpy, meaning the system released heat energy internally to maintain the adiabatic nature, leading to cooling.