During an isothermal compression of an ideal gas, 410 J of heat must be removed from the gas to maintain constant temperature. How much work is done by the gas during the process?

Verified step by step guidance

1. Identify the type of thermodynamic process: The problem states that it is an isothermal process (constant temperature) involving an ideal gas.
2. Recall the first law of thermodynamics, which is given by \(\Delta U = Q - W\), where \(\Delta U\) is the change in internal energy, \(Q\) is the heat added to the system, and \(W\) is the work done by the system.
3. Understand that for an ideal gas undergoing an isothermal process, the change in internal energy (\(\Delta U\)) is zero because internal energy is a function of temperature only, which remains constant.
4. Apply the first law of thermodynamics knowing \(\Delta U = 0\): \(0 = Q - W\). Solve for \(W\) to find the work done by the gas: \(W = Q\).
5. Substitute the given value of heat removed (\(Q = -410\) J, since heat is removed from the system, it is negative) into the equation to find the work done by the gas: \(W = -410\) J. The negative sign indicates that work is done on the gas.

Key Concepts

Isothermal Process

An isothermal process is a thermodynamic process in which the temperature of the system remains constant. For an ideal gas undergoing isothermal compression, the internal energy does not change, and any heat removed from the system is equal to the work done on the gas. This principle is crucial for understanding how energy is conserved in such processes.

First Law of Thermodynamics

The First Law of Thermodynamics states that energy cannot be created or destroyed, only transformed from one form to another. In the context of an isothermal process, this law can be expressed as ΔU = Q - W, where ΔU is the change in internal energy, Q is the heat added to the system, and W is the work done by the system. For isothermal processes involving ideal gases, ΔU is zero, simplifying the relationship to Q = W.

Work Done by a Gas

The work done by a gas during a thermodynamic process can be calculated using the formula W = PΔV, where P is the pressure and ΔV is the change in volume. In an isothermal compression, the gas does negative work on the surroundings as it is compressed, which corresponds to the heat removed from the system. Understanding this relationship is essential for calculating the work done in the given scenario.