Ideal gas equation
Ideal Gas Equation
The ideal gas equation is a fundamental equation in thermodynamics and physical chemistry that describes the behavior of an ideal gas. An ideal gas is a hypothetical gas that perfectly follows the kinetic molecular theory, with no intermolecular forces and where the molecules occupy no volume. Although no real gas behaves perfectly as an ideal gas, many gases at high temperature and low pressure approximate the behavior of an ideal gas quite well.
The Equation
The ideal gas equation is given by:
[ PV = nRT ]
where:
- ( P ) is the pressure of the gas,
- ( V ) is the volume of the gas,
- ( n ) is the number of moles of the gas,
- ( R ) is the universal gas constant, and
- ( T ) is the temperature of the gas in Kelvin.
The Universal Gas Constant
The universal gas constant ( R ) has different values depending on the units used for pressure and volume. Some common values are:
- ( R = 8.314 \, \text{J/mol·K} ) (when pressure is in Pascals and volume in cubic meters)
- ( R = 0.0821 \, \text{L·atm/mol·K} ) (when pressure is in atmospheres and volume in liters)
Derivation from Empirical Gas Laws
The ideal gas equation is derived from the combination of three empirical gas laws:
- Boyle's Law: At constant temperature, the pressure of a fixed amount of gas is inversely proportional to its volume (( P \propto \frac{1}{V} )).
- Charles's Law: At constant pressure, the volume of a fixed amount of gas is directly proportional to its temperature (( V \propto T )).
- Avogadro's Law: At constant temperature and pressure, the volume of a gas is directly proportional to the number of moles of gas (( V \propto n )).
Combining these laws gives us the ideal gas equation.
Differences and Important Points
| Property | Ideal Gas Behavior | Real Gas Behavior |
|---|---|---|
| Volume of Particles | Negligible compared to the container volume | Not negligible, especially at high pressure |
| Intermolecular Forces | None | Can be significant, especially at low temperature and high pressure |
| Compressibility | Highly compressible | Less compressible due to volume and forces |
| Temperature Range | Behaves ideally at high temperatures | Deviates from ideal behavior at low temperatures |
| Pressure Range | Behaves ideally at low pressures | Deviates from ideal behavior at high pressures |
Examples
Example 1: Calculating the Volume of a Gas
Suppose we have 2 moles of an ideal gas at a pressure of 1 atmosphere and a temperature of 273 K. We want to find the volume of the gas.
Using the ideal gas equation:
[ PV = nRT ]
We can solve for ( V ):
[ V = \frac{nRT}{P} ]
Substituting the values:
[ V = \frac{2 \, \text{mol} \times 0.0821 \, \text{L·atm/mol·K} \times 273 \, \text{K}}{1 \, \text{atm}} ]
[ V = 44.7 \, \text{L} ]
Example 2: Calculating the Pressure of a Gas
If we have 0.5 moles of an ideal gas in a 22.4 L container at a temperature of 0°C (273 K), what is the pressure of the gas?
Using the ideal gas equation:
[ P = \frac{nRT}{V} ]
Substituting the values:
[ P = \frac{0.5 \, \text{mol} \times 0.0821 \, \text{L·atm/mol·K} \times 273 \, \text{K}}{22.4 \, \text{L}} ]
[ P \approx 0.5 \, \text{atm} ]
Example 3: Calculating the Temperature of a Gas
If 1 mole of an ideal gas at a pressure of 2 atmospheres occupies a volume of 12.2 L, what is the temperature of the gas?
Using the ideal gas equation:
[ T = \frac{PV}{nR} ]
Substituting the values:
[ T = \frac{2 \, \text{atm} \times 12.2 \, \text{L}}{1 \, \text{mol} \times 0.0821 \, \text{L·atm/mol·K}} ]
[ T \approx 298 \, \text{K} ]
The ideal gas equation is a powerful tool in understanding the behavior of gases under various conditions. It is important to remember, however, that it is an approximation and that real gases can deviate from this behavior under certain conditions.