The ideal gas law relates pressure (|P|), volume (|V|), absolute temperature (|T|) and the amount of gas in moles (|n|) at a given moment.
The formula of the general gas law can be used to compare the characteristics of a gas in two different situations. However, this formula is not useful when you want to determine the characteristics of a gas at a specific point in time. The general gas law formula can be modified by introducing a proportionality constant. This constant, symbolised by the letter |R|, groups together all the constants of the simple gas laws. So, mathematically, the perfect gas law can be written as follows:
|PV=nRT|
where
|P| represents the pressure (in |\text{kPa}|)
|V| represents the volume (in |\text{L}|)
|n| represents the amount of gas (in |\text{mol}|)
|R| represents the perfect gas constant (in |\text{kPa}\cdot\text{L}/\text{mol}\cdot\text{K}|)
|T| represents the absolute temperature (in |\text{K}|)
The value of the perfect gas constant (|R|) can be determined using the value of the molar volume of a gas at TPN. Under these conditions, we find the following value:
|PV=nRT.|
Which can be rephrased as follows:
|R = \displaystyle \frac{P\times V}{n\times T}.|
Where the terms in the equation are replaced by the values at TPN:
|R = \displaystyle \frac{101{.}3\ \text{kPa}\times 22{.}4\ \text{L}}{1\ \text{mol}\times 273\ \text{K}}|
|R=8{.}314\ \text{kPa}cdot \text{L}/\text{mol}\cdot \text{K}.|
The ideal gas constant is equal to |8.314\ \text{kPa}\cdot \text{L}/\text{mol}\cdot \text{K}|. However, it is important that the units of measurement for the various characteristics are respected in order to be able to use this constant.
The ideal gas law describes the interdependence between the pressure, temperature, volume and number of moles of a gas at a given time. It can therefore be used to find an unknown variable when the other three are known.
What is the volume, in litres, occupied by |4\ \text{mol}| of methane, |\text{CH}_{4}|, at a temperature of |18\ \text{°C}| and a pressure of |1{.}4\ \text{atm}|?
- Identification of problem data or values
|P = {1{.}40\ \text{atm}}\times{101{.}3\ \text{kPa}}=142\ \text{kPa}|
|V = x|
|n = 4\ \text{mol}|
|R = 8{.}314\ \text{kPa}\cdot \text{L}/\text{mol} \cdot \text{K}|
|T = 18\ \text{°C} + 273{.}15 = 291{.}15\ \text{K}| - Volume calculation
|PV=nRT|
|V=\displaystyle \frac{nRT}{P}|
|V=\displaystyle \frac{{4}\times{8{.}314\ \text{kPa}\cdot\text{L}/\text{mol} \cdot \text{K}}\times{291{.}15\ \text{K}}}{142\ \text{kPa}}|
|V=68\ \text{L}|
Answer: The volume of methane is |68\ \text{L}|.
This law is based on the behaviour of a so-called ideal gas.
A ideal gas is a gas that theoretically complies with all the gas laws, regardless of temperature and pressure conditions, and whose properties can be explained by the kinetic theory of gases.
The particles in a perfect gas would therefore have the following characteristics:
- they do not interact with each other;
- they bounce back without losing energy;
- their collisions with obstacles are perfectly elastic;
- the gas does not liquefy, even at a temperature of |0\ \text{K}|.
In reality, however, there is no such thing as a perfect gas. In fact, under extreme conditions of temperature or pressure (far from TPN or TAPN), real gases cease to behave according to kinetic theory. Nevertheless, the ideal gas law can be used to study real gases under conditions similar to those at TPN and TAPN.
What is the mass of |\text{CO}_{2}| enclosed in a container of |3{.}5\ \text{L}| at a pressure of |101{.}6\ \text{kPa}| and a temperature of |26{.}3\ \text{°C}|?
- Identification of problem data or values
|P=101{.}6\ \text{kPa}|
|V=3{.}5\ \text{L}|
|n = ?|
|R=8{.}314\ \text{kPa}\cdot\text{L}/\text{mol}\cdot\text{K}|
|T = 26{.}3\ \text{°C} + 273{.}15 = 299{.}45\ \text{K}| - Calculation of number of moles
|PV=nRT|
|n=\displaystyle \frac{PV}{RT}|
|n=\displaystyle \frac{101{.}6\ \text{kPa}\times 3{.}5\ \text{L}}{8{.}314\ \text{kPa}\cdot\text{L}/\text{mol}\cdot\text{K} \times 299{.}45\ \text{K}}|
|n=0{.}14\ \text{mol}| - Calculation of the mass of |\text{CO}_{2}|
The molar mass (|M|) of |\text{CO}_{2}| is |44{.}01\ \text{g/mol}|.
|M=\displaystyle \frac{m}{n}|
|m=M\times n| |m=44{.}01\ \text{g/mol} \times 0{.}14\ \text{mol}|
|m=6{.}16\ \text{g}|
Answer: The mass of carbon dioxide is |6{.}16\ \text{g}|.