First Law Basics: Internal Energy, Work, Heat, Sign Convention

The first law of thermodynamics, also known as the law of energy conservation, asserts that energy cannot be created or destroyed, only transformed from one form to another. It emphasizes that the change in a system's internal energy is equal to the heat added to the system minus the work done by the system. This principle establishes a fundamental relationship between heat, work, and internal energy, highlighting the conservation of energy in thermodynamic processes.

When a sealed rigid container is heated and absorbs energy, the internal energy of the gas inside the container increases. Since the container is rigid, there is no change in volume, and therefore, no work is done by the gas (W = 0). According to the first law of thermodynamics, the change in internal energy (Δu) is equal to the heat absorbed (Q) plus the work done on the system. In this case, Δu = Q + W = 400 J + 0 J = +400 J. Thus, the internal energy increases by 400 J.

Internal energy is a state function, meaning it is determined solely by the state of a system, characterized by properties such as temperature, pressure, and volume. It does not depend on the path taken to reach that state. Therefore, the change in internal energy only relies on the initial and final states of the system, regardless of how the transition occurs. This property is fundamental in thermodynamics, as it allows for the calculation of energy changes in various processes without needing to consider the specific details of the process itself.

According to the first law of thermodynamics, the change in internal energy (Δu) of a system is given by the formula Δu = Q - W, where Q is the heat absorbed by the system and W is the work done by the system on its surroundings. In this case, the gas absorbs 500 J of heat (Q = +500 J) and does 200 J of work (W = +200 J). Substituting these values, Δu = 500 J - 200 J = +300 J, indicating an increase in internal energy.

Heat (q) and work (w) are not properties of a system but rather describe energy transfer during processes. Their values depend on the specific path taken between initial and final states, making them path-dependent. Unlike state functions, which depend only on the current state of a system, heat and work can vary with different processes, even if the initial and final states remain the same. Thus, they are classified as process quantities rather than state functions.

When a system releases heat, it is losing energy to the surroundings. In thermodynamics, the heat exchanged is represented by the symbol q. By convention, when heat is released from a system, it is considered to be negative because the system's internal energy decreases. This aligns with the first law of thermodynamics, which states that energy cannot be created or destroyed, only transferred. Thus, the release of heat signifies an exothermic process, resulting in a negative value for q.

In thermodynamics, the internal energy of a system increases when heat is added to it. According to the equation Δu = q - w, where Δu is the change in internal energy, q represents heat added to the system, and w is the work done by the system. If heat enters the system (q > 0) and no work is done (w = 0), the internal energy increases because the positive heat input contributes directly to the internal energy, while no energy is lost through work. Thus, this scenario leads to an increase in internal energy.

When a gas is compressed, work is done on the gas by the surroundings. In thermodynamics, work done by the system (the gas) is considered positive when it expands and negative when it is compressed. Since the gas is being compressed, it is losing energy to the surroundings, which results in negative work done by the gas. Thus, the work done by the gas in this scenario is negative.

In thermodynamics, an adiabatic process is characterized by the absence of heat transfer into or out of the system. When the heat transfer, represented by "q," is equal to zero (q=0), it indicates that the system is insulated from its surroundings, thus fulfilling the definition of an adiabatic process. Consequently, during an adiabatic process, any change in the internal energy of the system is solely due to work done on or by the system, confirming that the statement is true.

In the sign convention used in thermodynamics, work done by the system is considered positive because it indicates that the system is expending energy to perform work on its surroundings. This perspective helps in understanding energy transfer processes, where positive work signifies that the system is losing energy, while negative work indicates energy being added to the system. This convention is essential for analyzing thermodynamic cycles and ensuring consistent calculations across various processes.

To find the change in internal energy (Δu), we use the first law of thermodynamics, which states Δu = Q + W. Here, Q is the heat absorbed (250 J) and W is the work done on the gas (compression, which is negative, so W = -100 J). Substituting these values, we get Δu = 250 J + (-100 J) = 250 J - 100 J = 350 J. Thus, the change in internal energy is 350 J.

In an adiabatic process, no heat is exchanged with the surroundings. When a gas is rapidly compressed in an insulated cylinder, work is done on the gas, which increases its internal energy. This results in a rise in temperature as the gas molecules gain kinetic energy from the work applied. Unlike free expansion into a vacuum, where no work is done and internal energy remains constant, the rapid compression leads to a significant change in the system's energy state, demonstrating the principle that work done on a system can increase its internal energy.

When w (work done) equals zero, it indicates that the volume of the system remains constant throughout the process. This condition is known as an isochoric process. In an isochoric process, since there is no change in volume, no work is done on or by the system, and any heat added or removed results solely in a change in internal energy. This is significant in thermodynamics, as it helps in understanding the behavior of gases under constant volume conditions.

While Δu represents the change in internal energy of a system, it does not imply that temperature must remain constant. The internal energy can change due to heat transfer or work done on or by the system, which can occur at varying temperatures. For example, in an isothermal process, temperature remains constant, but in other processes, such as adiabatic or isochoric, temperature can change even if Δu is zero. Therefore, the statement is false.

To find the change in internal energy (Δu), we can use the first law of thermodynamics, which states that Δu = q + w. Here, q (heat added to the system) is -150 J and w (work done on the system) is +50 J. Plugging in these values: Δu = -150 J + 50 J = -100 J. However, since the answer provided is -200 J, it suggests a misunderstanding in the question or values. The correct calculation shows that Δu is actually -100 J, not -200 J, indicating a possible error in the options given.

In an adiabatic process, the internal energy change (Δu) of a gas is related to the work done on or by the gas. According to the first law of thermodynamics, Δu = Q - W, where Q is the heat added to the system and W is the work done by the system. Since the process is adiabatic, Q = 0. Therefore, Δu = -W. Given that the gas does 120 J of work, Δu = -120 J, indicating a decrease in internal energy.

In thermodynamics, the first law states that the change in internal energy (Δu) of a system is equal to the heat added to the system (q) minus the work done by the system (w). This relationship highlights the energy conservation principle, indicating that energy can be transferred as heat or work but is neither created nor destroyed. Hence, the equation captures how energy flows into and out of a system, with Δu representing the net change in internal energy resulting from these transfers.

Internal energy can change through heat transfer and work done on or by the system. Heat transfer occurs when energy moves between the system and its surroundings due to a temperature difference, affecting the internal energy. Similarly, when work is done on the system, energy is added, increasing internal energy, while work done by the system removes energy, decreasing it. Changing the system's mass without energy transfer does not contribute to internal energy changes, as it does not involve energy exchange. Thus, both heat transfer and work are essential processes for altering internal energy.

Internal energy of an object is the total energy contained within it, which includes various forms of energy at the molecular level. Random molecular kinetic energy refers to the energy due to the motion of molecules, while molecular potential energy arises from the interactions and bonds between them. Additionally, microscopic vibrations and rotations contribute to the internal energy as they represent the energy associated with the movement of molecules. Together, these forms of energy reflect the thermal state and behavior of the material at the microscopic scale.

A system's internal energy can increase due to work done on it, not only through heat absorption. According to the first law of thermodynamics, the change in internal energy is equal to the heat added to the system minus the work done by the system. Therefore, if work is done on the system, it can lead to an increase in internal energy even if no heat is absorbed. Thus, an increase in internal energy does not necessarily imply heat absorption.