The gas space above the water in a closed storage tank...

The gas space above the water in a closed storage tank contains nitrogen at $25^{\circ} \mathrm{C}$ and $100 \mathrm{kPa}$. Total tank volume is $4 \mathrm{m}^{3},$ and there is $500 \mathrm{kg}$ of water at $25^{\circ} \mathrm{C} .$ An additional $500 \mathrm{kg}$ of water is now forced into the tank. Assuming constant temperature throughout, find the final pressure of the nitrogen and the work done on the nitrogen in this process.

Key Concepts

Isothermal Process

An isothermal process is one that occurs at a constant temperature throughout the entire transformation. During this process, when a gas is compressed or expanded, its internal energy remains constant if we assume it behaves ideally. This implies that any work done on or by the gas is exactly balanced by heat exchanged with the surroundings, as dictated by the first law of thermodynamics.

Ideal Gas Law

The ideal gas law, expressed as PV = nRT, relates the pressure, volume, and temperature of an ideal gas. In problems such as the one given, this law is used to connect the change in the volume available to the gas with the corresponding change in its pressure, assuming the number of gas moles and the temperature remain constant. It serves as a fundamental tool in predicting the behavior of gases under different thermodynamic processes.

Thermodynamic Work in Gas Compression

Thermodynamic work, in the context of gas compression or expansion, is calculated as the integral of pressure with respect to volume change (W = ?P dV). For an isothermal reversible process involving an ideal gas, this leads to the expression W = nRT ln(V_final/V_initial), where the natural logarithm captures the relationship between the initial and final volumes. This concept is key to determining the energy transfer resulting from the compression or expansion work done on the gas.

Pressure-Volume (P-V) Relationship

The pressure-volume relationship describes how the pressure of a gas changes as its volume is altered during a thermodynamic process. Under constant temperature conditions, as seen in an isothermal process, the pressure and volume are inversely proportional; decreasing the volume forces an increase in pressure in accordance with Boyle’s Law. This relationship is essential for evaluating how changes in the system’s physical constraints affect the state of the gas.

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