Calculus assignment help

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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 distriubted 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 you calculations of the internal pressure, what can you say about the force of mutual attraction of molecules in an ideal gas?

1

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.