A 2-m^3 rigid tank initially contains air at 100 kPa and 22^∘ C . The tank is connected to a sup

step 1 In this question we have to find the fall in temperature of helium gas which is initially at 20

A 2-m^3 rigid tank initially contains air at 100 kPa and...

A $2-\mathrm{m}^{3}$ rigid tank initially contains air at $100 \mathrm{kPa}$ and $22^{\circ} \mathrm{C} .$ The tank is connected to a supply line through a valve. Air is flowing in the supply line at $600 \mathrm{kPa}$ and $22^{\circ} \mathrm{C}$. The valve is opened, and air is allowed to enter the tank until the pressure in the tank reaches the line pressure, at which point the valve is closed. A thermometer placed in the tank indicates that the air temperature at the final state is $77^{\circ} \mathrm{C}$. Determine (a) the mass of air that has entered the tank and $(b)$ the amount of heat transfer.

Key Concepts

Ideal Gas Law

The ideal gas law (pV = mRT) is a fundamental equation that relates pressure, volume, temperature, and mass of an ideal gas. This relation is used to determine the state properties of the gas at different conditions, such as calculating the mass of the gas in the tank when the pressure, volume, and temperature are known. It is applicable under the assumption that the gas behavior is close to ideal, which is a good approximation for air under many conditions.

Isochoric Process

An isochoric process is one that occurs at constant volume. In a rigid tank, the volume remains fixed, meaning that no work is done by or on the system during the process. This characteristic simplifies the analysis because any energy change in the system is solely due to heat transfer and internal energy change, rather than work interactions.

First Law of Thermodynamics for Closed Systems

The first law of thermodynamics states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system (?U = Q - W). In a constant volume (isochoric) process, since no work is done (W = 0), the heat transfer is equal to the change in internal energy (Q = ?U). This principle allows for the determination of the heat transfer by evaluating the change in the internal energy of the gas as its temperature changes.

Internal Energy and Specific Heats of Gases

For an ideal gas, the change in internal energy is directly related to its temperature change and the mass of the gas, typically using the specific heat at constant volume (c_v). The internal energy change is given by ?U = m * c_v * ?T, where ?T is the change in temperature. This relationship is essential for computing the heat transfer during the process, as it shows how variations in temperature affect the energy content of the gas.

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