A-level Chemistry/AQA/Module 5/Thermodynamics/Ideal Gas

What is an Ideal Gas
An Ideal Gas is a hypothetical gas in which all the molecules are considered as non-interacting elastic spheres with negligible radius. The molecules do not interact with each other and they move as free bodies except during very short time intervals when they bounce off each other or bounce off the wall that contains them. Obviously, there is no gas in nature that exhibits the characteristics of an Ideal Gas, although some come very close. It is only a crude approximation to the gases, which often gives reasonable results.

Why Do We Have The Ideal Gas Approximation
We use the ideal gas approximation in most thermodynamic examples involving gases. It is a very good approximation of real gases, especially at low pressures when particle interaction is low.

The Ideal Gas Temperature Scale
When defining a Temperature Scale, you must find a physical situation in which something measurable changes over time. The equation governing the change must be invertible. One option could be length of a metal rod, this wouldn't be very good though as the rod would soon melt.

What was settled on was the Ideal Gas Scale, Temperature is defined such that the product of Pressure and Volume is directly proportional to temperature. Such that 0 /Kelvin/ is when Pressure times Volume is 0 and 273.15K is the /Triple Point/ of Water. $$ PV \propto T $$

Equation of State of an Ideal Gas
Experimentally, it was realised that there were only a certain minimum number of properties of a substance which could be given any specified values. The others depended on these variables. The way in which they were related was called the Equation of State. The Equation of State of a an Ideal Gas is $$ \mathbf {PV = nRT} $$

P = Pressure

V = Volume

n = Number of Moles of Gas

R = Universal Gas Constant = 8.314 $$JK^-1M^-1$$

T = Temperature

The variables you set are called the independent variables, the others are called dependent.

Example
If you have 1 mole of Gas, and it occupies 1 metre cubed at 273.15K what pressure is it at?

Answer $$ \begin{matrix} \ PV & = & nRT \\ \\ \ P & = & \frac{nrt}{V} \\ \\ \ & = & \frac{1\cdot 8.314 \cdot 273.15}{1} \\ \\ \ & = & 2271 Pascals \end{matrix} $$

Pressure Volume Plots
A thermodynamic state of a gas (ideal or not) is completely described by four variables $$P$$, $$V$$, $$T$$ and $$n$$ (see previous topics). In most cases $$n$$ (the number of mols) is kept constant or is approximately constant (in other words the number of the gas molecules is practically unchanged) so we are left with three variables. From the equation of state (see previous topics) we can calculate $$T$$ from the other two variables. Therefore we are left with only 2 independent variables $$P$$ and $$V$$. It is common practice to represent a thermodynamic state on a $$P$$-$$V$$ diagram with a point. The coordinates of the point give us the values of $$P$$ and $$V$$. A set of such points represent a thermodynamic process, as shown for example in the figure below.



The process a) is a constant pressure prosess, known as isobaric (from the Greek word iso:equal and baros:load). For example the steam that leaves our body after a shower. It keeps increasing its volume (it is to be found all over the bathroom) but since it is in open air it has a constant atmospheric pressure. The process b) is a constant volume prosess, known as isochorus (from the Greek word iso:equal and choros:space). For example when we put in the microwave oven a sealed food container. The traped air inside it occupies a costant volume but its pressure keeps rising as it warms up (that's why the lids sometimes pop-up in the microwave oven). The process c) is a constant temperature prosess, known as isothermal (from the Greek word iso:equal and thermal:heat). Why is it that the isothermal curve has such a shape? From the equation of state if $$T$$ = constant then $$PV$$ = constant and so $$P$$ $$\propto$$ $$1/V$$.