Question

A gas is enclosed in a cylinder with a movable frictionless piston. Its initial thermodynamic state at pressure \(P_{i}=10^{5} \mathrm{~Pa}\) and volume \(V_{i}=10^{-3} \mathrm{~m}^{3}\) changes to a final state at \(P_{f}=(1 / 32) \times 10^{5} \mathrm{~Pa}\) and \(V_{f}=8 \times 10^{-3} \mathrm{~m}^{3}\) in an adiabatic quasi-static process, such that \(P^{3} V^{5}=\) constant. Consider another thermodynamic process that brings the system from the same initial state to the same final state in two steps: an isobaric expansion at \(P_{i}\) followed by an isochoric (isovolumetric) process at volume \(V_{f} .\) The amount of heat supplied to the system in the two-step process is approximately (A) \(112 \mathrm{~J}\) (B) \(294 \mathrm{~J}\) (C) \(588 \mathrm{~J}\) (D) \(813 \mathrm{~J}\)

Short answer

The amount of heat supplied to the system in the two-step process is approximately 294 J.

Step-by-step solution

- Calculate Work Done in Isobaric Expansion

For an isobaric process, the work done on the system is given by the formula: \( W = P \times \triangle V \) where \( P \) is the pressure and \( \triangle V \) is the change in volume. Calculate the work done during the isobaric expansion at pressure \( P_{i} \) from \( V_{i} \) to \( V_{f} \): \( W_{\text{isobaric}} = P_{i} \times (V_{f} - V_{i}) \).

- Calculate Heat Supplied in Isobaric Expansion

For an isobaric process, the heat supplied to the system is equal to the work done, as there is no change in internal energy at constant pressure. Therefore, \( Q_{\text{isobaric}} = W_{\text{isobaric}} \).

- Determine the Change in Internal Energy in Isochoric Process

In an isochoric process, no work is done as there is no change in volume. However, the change in internal energy (\( \triangle U \)) can be calculated using the first law of thermodynamics, \( \triangle U = Q - W \). Since \( W = 0 \) for an isochoric process, \( \triangle U = Q_{\text{isochoric}} \).

- Calculate the Final Pressure After the Isobaric Expansion

The isobaric expansion proceeds at constant pressure \( P_{i} \), so the pressure after the expansion, before the isochoric process, is still \( P_{i} \).

- Apply the Adiabatic Condition to Find Final Pressure

Using the adiabatic condition \( P^3 V^5 = \) constant, we apply initial and final states to find \( P_{f}^3 V_{f}^5 = P_{i}^3 V_{i}^5 \). This gives us \( P_i = \left( \frac{P_f^3 V_f^5}{V_i^5} \right)^{1/3} \).

- Calculate the Change in Internal Energy Using the Ideal Gas Law for Isochoric Process

Knowing the initial and final pressures and the volume after isobaric expansion, use the ideal gas law \( P V = nRT \) to find the number of moles \( n \) and the temperature change. With the molar heat capacity at constant volume \( C_V \) and \( \triangle T \), calculate \( \triangle U = n C_V \triangle T \) for the isochoric process. For monatomic ideal gas, \( C_V = \frac{3}{2}R \).

- Calculate Heat Supplied in Isochoric Process

Because no work is done in an isochoric process, all supplied heat goes into changing the internal energy. Therefore, \( Q_{\text{isochoric}} = \triangle U \) from Step 6.

- Calculate Total Heat Supplied in Two-Step Process

Sum the heat supplied in the isobaric (Step 2) and isochoric (Step 7) processes to get the total heat supplied: \( Q_{\text{total}} = Q_{\text{isobaric}} + Q_{\text{isochoric}} \).

Key concepts

Isobaric Expansion

Isobaric expansion is a thermodynamic process where a gas expands at a constant pressure. In the context of our textbook problem, the gas in the cylinder expands under constant pressure, pushing the piston outward and doing work on the surrounding environment. The work done by the gas during isobaric expansion can be calculated using the formula
\( W = P \times \Delta V \)
where \( P \) is the constant pressure and \( \Delta V \) is the change in volume.

One unique aspect of an isobaric process is that the heat supplied to the system (\( Q_{\text{isobaric}} \) is directly converted into work with no change in internal energy, according to the first law of thermodynamics. This relationship is critical for understanding how energy transfers occur under constant pressure conditions in thermodynamic systems.

In our exercise, the initial volume \( V_{i} \) increases to \( V_{f} \) while the pressure remains at \( P_{i} \) during the expansion step. Calculating the work done provides us with the amount of heat transferred to the gas, as they are equivalent in this scenario.

Isochoric Process

In contrast to isobaric expansion, an isochoric process, also known as an isovolumetric process, takes place at a constant volume. No work is done since the volume of the system does not change, which can be represented mathematically as \( W = 0 \) because the work done is a product of pressure and change in volume, and here the change in volume is zero.

During an isochoric process, any heat added or removed from the system will result in a change in the internal energy, which can be described by the formula \( \Delta U = Q - W \). Since \( W = 0 \) for an isochoric process, all the heat supplied translates into a change in internal energy, or \( \Delta U = Q_{\text{isochoric}} \) for the gas.

For the two-step process in our textbook problem, after the gas undergoes an isobaric expansion, it then undergoes an isochoric process until it reaches the final state. This means that we need to calculate the internal energy change to find the heat supplied during this phase, as it doesn't perform any mechanical work.

Adiabatic Process

An adiabatic process is one where no heat is exchanged between the system and its surroundings. This implies a thermally insulated system, making the process inherently different from isobaric and isochoric processes where heat transfer occurs. In an adiabatic process, changes in the internal energy of the system result in work done by or on the system, affecting its temperature and pressure.

The relationship for an adiabatic process often involves specific heat capacities and can be described by a unique equation, such as \( P^3 V^5 = \text{constant} \) in the textbook exercise. This equation is derived from the general adiabatic condition \( PV^\gamma = \text{constant} \) where \( \gamma \) is the heat capacity ratio (\(\gammaeq1\)), but with the powers tailored to the particular characteristics of the process in question.

Using this adiabatic condition, one can relate the initial and final states of the system to determine properties like the final pressure or volume without considering heat transfer. The exercise leverages this condition to compare the more complex adiabatic process to a simpler two-step process involving isobaric and isochoric steps.