Calculus Lab 4: Internal Versus External Gas Pressure

Namecalculus Lab 4internal Versus External Pressure Of A Gasif The Vol

Namecalculus Lab 4internal Versus External Pressure Of A Gasif The Vol

Name Calculus Lab 4 Internal Versus External Pressure of a Gas If the volume of a gas is fixed and its temperature is permitted to vary, then the external pressure will, presumably, also need to vary according to the terms of the equation of state of the gas. External pressure is the pressure exerted by the gas on the walls of its container. This pressure is denoted by p(T ) or just by the letter p. The rate of change of external pressure with respect to time, p′(T ), is needed in the formula for the internal pressure of the gas. The internal pressure of a gas gives us information about the attractive force between pairs of individual molecules of the gas, a force that tends to reduce the external pressure.

This force is explained by the mutual attraction of molecules whose positive and negative charges are asymmetrically distributed in the geometry of the molecule. Water is an important example of such an asymmetry. For any gas, the internal pressure can be computed using the difference, T ∗ p′(T )− p(T ). The first term in the difference, T ∗ p′(T ) is called the latent heat of volume expansion, denoted Lv.

1. Take an ideal gas with molar equation of state p(T ) = RT v. R is the ideal gas constant. If the volume, v, is fixed, then p′(T ) = . Now, using the difference formula above calculate the internal pressure of an ideal gas with fixed volume. Based on your calculations of the internal pressure, what can you say about the force of mutual attraction of molecules in an ideal gas?

2. The equation of state for a van der Waals gas is p(T ) = RT v − b − a v2 , where R, a, and b are constants. Now fix the volume and calculate the internal pressure of a van der Waals gas. Based on your calculation, what can be said about the meaning of the term a v2 that is in the formula for the equation of state for a van der Waals gas?

3. The equation of state for a Bertholot gas is p(T ) = RT v − b − a Tv2 where R, a, and b are constants. Fix the volume and calculate the internal pressure of a Bertholot gas.

4. The equation of state for a Redlich-Kwong gas is p(T ) = RT v − b − a v(v − b) √ T where R, a, and b are constants. Fix the volume and calculate the internal pressure of a Bertholot gas.

Paper For Above instruction

The concepts of internal and external pressure in gases are fundamental to understanding thermodynamic behavior. The relationship between these pressures, especially under different models such as ideal gases, van der Waals gases, Bertholot, and Redlich-Kwong gases, illuminates the microscopic interactions and the forces at play between molecules. The interplay of these pressures not only explains the macroscopic properties of gases but also provides insights into molecular attractions and repulsions that define real gases' behavior.

For an ideal gas, the internal and external pressures are essentially equivalent under ideal conditions. According to the ideal gas law, p(T) = RT/v, where R is the universal gas constant, T is temperature, and v is molar volume. The derivative p′(T) = 0 because the pressure of an ideal gas does not depend on temperature in a manner that changes with T linearly for fixed volume. Applying the given difference formula, T p′(T) - p(T), results in zero, indicating no internal pressure—reflecting the absence of intermolecular attractive forces in ideal gases. This aligns with the kinetic molecular theory, which assumes particles do not exert forces on each other, and collisions are perfectly elastic.

In realistic scenarios, however, gases exhibit intermolecular forces. Van der Waals' model modifies the ideal gas law by incorporating repulsive and attractive forces through parameters a and b. The term a/v² indeed signifies the strength of attractive forces among molecules; larger values of a imply stronger attraction, which reduces the overall pressure compared to an ideal gas at the same temperature and volume. Applying the differential approach, the internal pressure for a van der Waals gas becomes dependent on these parameters, providing a measure of how molecular interactions influence the thermodynamic state.

Similarly, the Bertholot gas introduces further modifications to account for molecular interactions with a different theoretical approach, reflected in its specific equation of state p(T) = RT/v - b - a Tv². Fixing the volume and calculating the internal pressure using the difference formula allows us to analyze the effect of these additional molecular interactions. The parameters a and b in all models effectively quantify the extent to which intermolecular forces impact the pressure and thermodynamic behavior of gases.

The Redlich-Kwong model, which incorporates a temperature-dependent term in the attractive forces, offers a more accurate description of real gases over a wider range of conditions. Its equation, p(T) = RT/v - b - a v(v - b) / √T, emphasizes the temperature dependence of intermolecular attractions. Calculating the internal pressure using this model, with a fixed volume, highlights the importance of temperature-dependent forces in real-gas behavior.

In conclusion, the internal pressure calculations across different models reinforce the understanding that intermolecular forces significantly influence the thermodynamic properties of gases. While ideal gases assume no interactions, real gases exhibit attractive forces that are modeled via parameters like a in van der Waals' and other extended equations of state. These parameters provide valuable insights into the molecular attractions and their impact on the macroscopic behavior of gases, which is essential for applications across physics, chemistry, and engineering disciplines.

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