Question

The isothermal compressibility factor ‘ $\alpha $ ‘ for an ideal gas at 0.25 atm pressure is?

Answer 1

Hint: The isothermal gas compressibility, ${c_g}$, is a useful concept that is used extensively in determining the compressibility properties of the reservoir. Gas usually is the most compressible medium in the reservoir. The formula used to determine the isothermal compressibility factor ‘ $\alpha $ ‘ is $\dfrac{1}{P}$ where $P$ is the pressure.

Complete step by step answer:
Step 1:
Gas compressibility or it is also called the coefficient of isothermal gas compressibility and also called the coefficient of isothermal compressibility of the gas. It is defined as the relative change in the volume of the gas with respect to change in pressure at constant pressure. It is expressed in units of reciprocal pressure. The unit of pressure is a bar or atm and the reciprocal of the pressure is usually called $ps{i^{ - 1}}$.

Step 2:
The isothermal gas compressibility is defined by the following equation:
Here we have taken it for real gas having temperature constant ${c_g}$= $ - \dfrac{1}{V}{\left( {\dfrac{{dV}}{{dp}}} \right)_T}$.
The compressibility of an ideal gas is given by the following equation ${c_g}$= $\dfrac{1}{P}$ where P is the pressure.
For real gas, compressibility is given by the following equation: ${c_g}$= \[\dfrac{1}{P} - \dfrac{1}{z}{\left( {\dfrac{{dz}}{{dp}}} \right)_T}\].
At low pressures, when gas behaves as if it were an ideal gas, the z-factor derivative with respect to pressure equals to zero, and the gas compressibility gives the same form as the ideal gas compressibility.

Step 3:
The isothermal compressibility for gases at low pressure is known by a formula.
It is denoted as ${c_g}$= $\dfrac{1}{P}$ where $P$ is the pressure
We are given pressure equals to 0.25 atm (atmospheric pressure)
Substituting into the formula we get $\dfrac{1}{{0.25}}$
$=4$

$\therefore $ The value of $\alpha = 4$.

Note:
The compressibility factor is useful in the thermodynamic property for modifying the ideal gas law to account for the behavior of real gases. It is a measure of how much the thermodynamic properties of a real gas deviate from those expected of an ideal gas.
Uses :
- On compressing the petroleum gas it forms a liquid.
- Oxygen cylinders used for various purposes in hospitals, divers, and many other works.
- CNG(compressed natural gas) is natural gas.
- Methane, which is compressed and used as a fuel in vehicles and at homes too.