Question

A system goes from state 1 to state 2 and back to state 1 . (a) What is the relationship between the value of $\mathrm{\Delta }E$ for going from state 1 to state 2 to that for going from state 2 back to state 1 ? (b) Without further information, can you conclude anything about the amount of heat transferred to the system as it goes from state 1 to state 2 as compared to that upon going from state 2 back to state $1?$ (c) Suppose the changes in state are reversible processes. Can you conclude any thing about the work done by the system upon going from state 1 to state 2 as compared to that upon going from state 2 back to state $1?$

Short answer

In conclusion: (a) The change in internal energy for going from state 1 to state 2, ΔE(1-2), is equal and opposite to the change in internal energy for going from state 2 back to state 1, ΔE(2-1), i.e. ΔE(1-2) = -ΔE(2-1). (b) We cannot make any definitive conclusion about the heat transfer in both processes without further information. (c) If the processes are reversible, the work done by the system during both processes will be equal but with opposite signs, i.e., W(1-2) = -W(2-1).

Step-by-step solution

Understanding the System and Processes

We are given a system that goes from state 1 to state 2 and then back to state 1. Let's denote the change in internal energy for processes 1 -> 2 and 2 -> 1 as ΔE(1-2) and ΔE(2-1), respectively.

Internal Energy changes for processes 1 -> 2 and 2 -> 1

By the first law of thermodynamics, ΔE = Q - W where ΔE is the change in internal energy, Q is the heat transferred to the system, and W is the work done by the system. Since the system returns to state 1 after going to state 2, the initial and final states are the same. Hence, the changes in internal energy for both processes should be opposite because the net change in internal energy must be zero. Mathematically, this can be represented as: ΔE(1-2) = -ΔE(2-1)

Analyzing the heat transfer to the system

We are not given any specific information about the processes involved in going from state 1 to state 2 or from state 2 to state 1. Therefore, without any further information, we cannot determine the relationship between the heat transfer to the system during both processes.

Analyzing the work done by the system during reversible processes

Assuming that the processes from state 1 to state 2 and from state 2 back to state 1 are both reversible, we can say that the work done by the system during both processes should be equal but with opposite signs. Mathematically, this can be represented as: W(1-2) = -W(2-1) This is because reversible processes are characterized by being able to return a system to its original state without any change in the surroundings. So, the work done by the system during the process of going from state 1 to state 2 should be equal (but opposite in sign) to the work done during the process of going from state 2 back to state 1. In conclusion, for the given system, the change in internal energy, ΔE(1-2), is equal and opposite to the change in internal energy, ΔE(2-1). We cannot make any definitive conclusion about the heat transfer in both processes without further information. However, if the processes are reversible, the work done by the system during both processes will be equal but with opposite signs.

Key concepts

Internal Energy

Internal energy ($E$ ) is a fundamental concept in thermodynamics referring to the energy contained within a system due to the specific configurations of its molecules. This includes energies related to molecular vibrations, rotations, and other potential and kinetic energies at the molecular level.
When we talk about changes in internal energy ($\mathrm{\Delta }E$), we are focusing on the difference in this contained energy between two states of the system.

• If a system undergoes a process, perhaps from state 1 to state 2 and then back to state 1, the change in internal energy over each path can be calculated.
• In reversible cycles, the overall change in internal energy should be zero, which implies that any gain in internal energy ($\mathrm{\Delta }E(1-2)$) while going from state 1 to state 2 is exactly offset by a loss ($\mathrm{\Delta }E(2-1)$) when returning from state 2 back to state 1.

Internal energy is not directly measureable, but changes in internal energy can be calculated using the equation derived from the first law of thermodynamics.

First Law of Thermodynamics

The first law of thermodynamics is a central principle in physics, sometimes referred to as the law of energy conservation. It states that energy can neither be created nor destroyed, only transformed from one form to another or transferred from one system to another.
In equation form, this law is given by:$$\mathrm{\Delta }E=Q-W$$where:

• $\mathrm{\Delta }E$: Change in internal energy,
• $Q$: Heat added to the system,
• $W$: Work done by the system.

This equation reflects that the change in a system's internal energy is the net result of energy added through heat minus the energy used for work.
For cyclic processes, such as the one described where the system returns to its original state, the total change in internal energy should be zero, reinforcing that the work done over one full cycle ($W_{cycle}$) and the heat transferred ($Q_{cycle}$) must balance each other.

Reversible Processes

Reversible processes are a special type of thermodynamic process where the system, along with its surroundings, can be restored to their initial states with no net changes. These processes are theoretical constructs that are used to represent ideal scenarios.

• In reversible processes, the path taken between two states is well-defined and any deviations would result in a loss of reversibility, potentially creating wastage or increasing the entropy of the system.
• Key characteristics involve infinitesimally small gradients between the system and its surroundings, allowing for gradual energy transfers.

Practically, no process is perfectly reversible due to inevitable losses such as friction, but reversible processes provide a benchmark for maximum efficiency.
For the given problem, if the transitions between states 1 to 2 and 2 back to 1 are reversible, the work done merely reverses direction. It stays equal in magnitude but changes sign to reflect this back-and-forth motion, i.e., $W(1-2)=-W(2-1)$.This characteristic guides the understanding of energy transactions in idealized thermodynamic systems.