Ideal Gas Laws

Background on Ideal Gases

Robert Boyle made the first quantitative measurements of gases in a systematic manner. Using a manometer, which measures differences in pressure, and a barometer, which measures the total pressure of the atmosphere, he developed what is now known as Boyle's law. This law states that at any constant temperature, the product of the pressure and the volume of any size sample of any gas is a constant, PV is constant when T is constant.

Several years later, French chemist Jacques Charles formulated a general law known as Charles's law. This law states that at any constant pressure, the volume of any sample of any gas is directly proportional to the temperature, V/T is constant when P is constant.

Discovered in 1802, the third gas law, Gay-Lussac's law, states that the pressure of a fixed amount of gas at fixed volume is directly proportional to its temperature in kelvins. This law was named after the French chemist Joseph Louis Gay-Lussac. It is expressed mathematically as P/T is constant when V is constant.

Example of the three gas laws, here. Keep T const. for Boyle's law, P const. for Charles' law, and V const. for Gay-Lussac's law.

So we've heard of all these variables, but what exactly are they?

• Pressure
• Temperature
• Volume/Density

The Ideal Gas Law

The three laws we just talked about describe three fundamental relationships between, P and T, V and T, and V and P. Now, can combine the laws to make the combined gas law:

       P V = constant
        T

For atmospheric science, we don't want to think about a gas in terms of volume, it is hard to calculate the volume of atmosphere over a given region. We want to think about gases in terms of density, mass per unit volume. This is much easier to calculate. Using density instead of volume gives us the common form of the Ideal Gas Law in atmospheric science:
                
Where...

• p is pressure in Pascals (Pa).
• &rho is density in kg/m^3.
• T is temperature in Kelvin (K).
• R is a constant, its value changes with each gas.
           For dry air , R = 287 J/kg K.

         You will need Java for this application, www.java.com

This applet allows you to control the
         &rho (number of molecules and volume), P and T.
• What happens when you increase the density, by adding more molecules of the same temperature to a constant volume?
  
  • The pressure increases
• What happens when you increase the density, by changing the volume?
  
  • Both pressure and temperature increases.
• What happens when you increase the temperature, at constant pressure?
  
  • The density decreases, because V increases.
• What happens when you increase the temperature, at constant volume?
  
  • The pressure increases as the molecules move faster.


Play around with this applet to discover the Ideal Gas Law relationships on your own.

Let's try a practice problem

If the temperature of an air parcel is -20.5 C, and its density is 0.690 kg/m^3, what is the pressure of the air parcel, in mb?

P = &rho R T

T = -20.5 C + 273 K = 252.5 K
&rho = 0.690 kg/m^3
R = 287 J/kg K

So, p = 0.690 kg/m^3 * 252.5 K * 287 J/kg K = 50000 Pa
In millibars, p ~ 50,000 Pa / 100(Pa/mb) = 500 mb

Temperature and Heat

Heat capacity is the amount of heat required per unit increase in temperature. It is a measure of how well the substance stores heat, each substance has its own heat capacity

H. C. = (heat added) / (change in temperature)
• Materials with large heat capacities hold heat well.
  Their temperatures will not rise much for a given amount of heat.
  Example: water, air
• Materials with small heat capacities do not hold heat well.
  Their temperatures will rise quickly for a given amount of heat.
  Example: copper, soil

Specific Heat = (Heat capacity) / (mass), or c.
More specifically, c is the amount of energy required to raise the temperature of one kilogram of any substance by 1 degree Kelvin.

Specific heat is measured in Joules per kilogram Kelvin, J/(kg K)
Substances with different specific heats require different amounts of energy.

• Low specific heat: Less input energy to raise the temperature by 1 degree Kelvin.
  • Example: All metals
• High specific heat: High input energy to raise the temperature by 1 degree Kelvin.
  • Example: Water

First Law of Thermodynamics

The first law of thermodynamics is the application of the conservation of energy principle to heat and thermodynamic processes:

Where U is the internal energy, Q is the heat added to the system, and W is work done by the system.

So what is:

Internal energy?
Work done by a system?

To start: push the pressure constant button, then press the reset button. The volume will start at the smallest size. Now, type in 100 molecules in the number of molecules section.
• You will see that the volume increases, so &Delta V is positive. This means that the gas is doing work to expand the volume. Work done by the system is positive.
• You will also notice that the temperature increases. This means that the kinetic energy is increasing, therefore the internal energy is increasing.

Based on the internal energy and work, is heat being added to the system or removed from the system?