Question

One \(\mathrm{kmol}\) of argon at \(300 \mathrm{~K}\) is initially confined to one side of a rigid, insulated container divided into equal volumes of \(0.2 \mathrm{~m}^{3}\) by a partition. The other side is initially evacuated. The partition is removed and the argon expands to fill the entire container. Using the van der Waals equation of state, determine the final temperature of the argon, in \(\mathrm{K}\). Repeat using the ideal gas equation of state.

Short answer

The final temperature is 300 K for both the van der Waals and ideal gas equations.

Step-by-step solution

- Understanding the van der Waals Equation

The van der Waals equation is given by \[ \left( P + \frac{a}{V_m^2} \right)(V_m - b) = RT \]where:- \(P\) is the pressure,- \(V_m\) is the molar volume,- \(a\) and \(b\) are van der Waals constants,- \(R\) is the gas constant,- \(T\) is the temperature.

- Calculate Initial Molar Volume

Initially, argon is confined to one side of the container with volume 0.2 m³. The initial molar volume: \[ V_{m1} = \frac{0.2 \mathrm{~m}^3}{n} \]Given that \( n = 1 \mathrm{~kmol} \),\[ V_{m1} = 0.2 \mathrm{~m}^3 \]

- Calculate Final Molar Volume

After the partition is removed, the volume available doubles to 0.4 m³. Thus,\[ V_{m2} = \frac{0.4 \mathrm{~m}^3}{1 \mathrm{~kmol}} = 0.4 \mathrm{~m}^3 \]

- Applying Energy Conservation (Isothermal Process)

Since the container is insulated and rigid, no heat exchange or work is done. For van der Waals gas under these conditions, an isothermal process can be assumed:\[ T_{final} = T_{initial} \]Thus, for the van der Waals gas,\[ T_{final} = 300 \mathrm{~K} \]

- Using Ideal Gas Law to Verify

The ideal gas equation is given by \[ PV = nRT \]For an isothermal expansion, the initial and final temperatures remain the same:\[ T_{final} = T_{initial} = 300 \mathrm{~K} \]

Key concepts

thermodynamic processes

Thermodynamic processes describe how a system's state changes over time by exchanging energy or matter with its surroundings. The system can undergo processes like expansion, compression, heating, cooling, or chemical reactions. In the given exercise, the argon undergoes an **isothermal expansion**. This means the temperature remains constant throughout the process. Here, the gas expands into a vacuum, so no external work is done, and no heat is exchanged due to the container being insulated.
The key takeaway is that the system's state change needs to be consistent with energy conservation principles.
Understanding different types of thermodynamic processes helps in predicting the final state of systems efficiently.

ideal gas law

The Ideal Gas Law is one of the most fundamental concepts in thermodynamics and chemistry. It’s represented by the equation: \( PV = nRT \). Here:

• \( P \) is pressure
• \( V \) is volume
• \( n \) is number of moles
• \( R \) is the gas constant
• \( T \) is temperature

The law provides a good approximation for the behavior of gases under a wide range of conditions, although it assumes that the gas particles do not interact and occupy no volume themselves.
In the exercise, the ideal gas law helps verify that the final temperature remains the same as the initial temperature after an isothermal expansion. Remember that this law provides a simplified model that is most accurate when intermolecular forces are negligible.

molar volume

Molar volume is another concept that plays a significant role in understanding how gases behave. It is the volume occupied by one mole of a substance. The formula to calculate molar volume is: \( V_m = \frac{V}{n} \). Here:

• \( V \) is the volume
• \( n \) is the number of moles

In the exercise:

• The initial molar volume is calculated for a volume of 0.2 m³
• The final molar volume doubles to 0.4 m³ after the partition is removed

Understanding molar volume helps in determining how the gas expands and fills the container. It's crucial for intricate calculations of gas properties and behaviors.

isothermal expansion

Isothermal expansion is a specific type of thermodynamic process where the temperature remains constant while a gas expands. This can happen when the system is in thermal equilibrium with a heat reservoir that absorbs or supplies heat as necessary to maintain a constant temperature.
In the given exercise, the container is thermally insulated and rigid. This ensures no heat transfer or work done on the surroundings, allowing us to assume an isothermal expansion accurately. The key point is:

• The final temperature of the gas remains the same as the initial temperature, i.e., 300 K

.Understanding isothermal expansion helps in predicting changes in volume and pressure of a gas under steady temperature conditions.