The relationship between volume and temperature was determined much later because accurate thermometers had to be developed first. But once thermometers were available a number of workers determined that volume is directly proportional to temperature. Plotting data for the relation of volume of a gas to temperature between 0° C and 100 ° C gives a plot similar to that below:
Extrapolating this data to V = 0 we can find an absolute minimum value of temperature on the assumption that negative volumes can't exist:
The intercept on the volume axis is then taken as absolute zero = -273.15 °C = 0 K for an ideal or "perfect" gas with particles of zero volume and no interactions other than collisions.
Algebraically we then find that V = k'T, & V1/T1 = V2/T2.
We can combine Boyle's and Charles' relationships (T was part of the constant for Boyle's Law and P is part of the constant for Charles' Law) to give:
V = an, where n = moles of stuff. So we have a linear relation between volume and moles; and V1/n1 = V2/n2.
Ideal Gas Law ("Perfect Gas Law"): The constant for the combined law includes amount of stuff, and breaking that out we then get
where R = the gas constant with units appropriate to the various measurements. We will use atm, L, K, and moles, so that
I will base all of my examples on this equation because that requires a minimum of memorization. However you may find it easier to memorize a series of equations such as the "combined gas law equation" etc.
As an example, let's find the density of sulfur hexafluoride at a temperature of 25 °C and a pressure of 767 mmHg. The first thing we should do is determine what we want to know, mainly the mass of gas in a one Liter sample.
To do this we need to find 1) moles/L and 2) grams/mole. So let's write out the Ideal Gas Law:
Rearranging to find moles/L,
n/V = P/RT
Next let's list what we know in this equation:
- P = 767 mmHg
- V = 1.000 L (assume one liter since want density in mol/L)
- n = ?
- R = 0.0821 L*atm/mole*K
- T = 25°C
Looking at these values we see that we need to convert P to atm and T to K in order to match the units in R!
- P = (767 mmHg)(1 atm/760 mmHg) = 1.009 atm
- T = 25 + 273.15 K = 298.15 K
Plugging these values into the gas law equation
n/V = P/RT = (1.009atm)/(0.0821L*atm/mole*K)(298.15K)
n/L = 4.122 x10-2 moles/one liter
The MW of SF6 = [32.07 + 6(19.00)] g/mol = 146.07 g/mol. Multiplying mol/L by g/mol then gives the density, D:
D = (4.122 x10-2 mol/L)(146.07 g/mol) = 6.0210 g/L = 6.02 g/L
Example: In an experiment the pressure of nitrogen inside a sealed, rigid container is found to be 15.6 atm at 557 °C. What was the pressure in the container when it was sealed in the lab at 25°C?
First look at the problem to see what changes and what doesn't.
- R is a constant so doesn't change.
- If the can is sealed, then n is constant.
- If it doesn't expand (its rigid), then V is constant.
Next write out the gas law equation,
PV = nRT
Rearranging to put the variables on one side:
P/T = nR/V, and since nR/V is constant, Po/To = Pf/Tf
Rearranging: Pf = (Po)(Tf/To)
Next, convert temperatures into Kelvins
- T1 = 25 + 273.15 = 298.15 K
- T2 = 557 + 273.15 = 830.15 K
Pf = (15.6 atm)(298.15 K)/(830.15 K)
= 5.133 atm = 5.13 atm.
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© R A Paselk
Last modified 18 February 2011