11.5: Universal Gas Law
Compressed gases provide vital fuels for industry and for homes and farms in rural areas.
Universal Gas Law
The combined gas law, \begin{align*}PV \propto T\end{align*}
Because different gases have different weights per molecule, including a term for mass of gas does not produce a consistent equation. If, however, we include a term expressing the number of moles of gas rather than its mass, we can produce a constant proportionality. A mole is a unit representing the number of atoms present. The letter \begin{align*}n\end{align*}
The unit term for \begin{align*}n\end{align*}
Pressure Units | Volume Units |
Units for \begin{align*}\underline{n}\end{align*} |
Units for \begin{align*}\underline{T}\end{align*} |
Value of \begin{align*}\underline{R}\end{align*} |
atm | liters | moles | Kelvin | 0.0821 L•atm/mol•K |
atm | milliliters | moles | Kelvin | 82.1 mL•atm/mol•K |
Since the product of (liters)(atm) can be converted to joules, we also have a value for \begin{align*}R\end{align*}
Most universal gas law problems are calculated at STP. STP stands for standard temperature and pressure, which is the most commonly calculated temperature and pressure value. STP is defined as 1.00 atm and 0°C, or 273 K.
Example Problem: Determine the volume of 1.00 mol of any gas at STP.
Solution: First isolate V from PV=nRT. Then plug in known values and solve.
\begin{align*}V=\frac{nRT}{P}=\frac{(1.00 \ \text{mol})(0.0821 \ \text{L} \cdot \text{atm/mol} \cdot \text{K})(273 \ \text{K})}{(1.00 \ \text{atm})}=22.4 \ \text{liters}\end{align*}
For any gas at STP, one mole has a volume of 22.4 liters. This can be an extremely convenient conversion factor.
Example Problem: A sample of oxygen gas occupies 10.0 liters at STP. How many moles of oxygen are in the container?
Solution:
\begin{align*}n=\frac{PV}{RT}=\frac{(1.00 \ \text{atm})(10.0 \ \text{L})}{(0.0821 \ \text{L} \cdot \text{atm/mol} \cdot \text{K})(273 \ \text{K})}=0.446 \ \text{moles}\end{align*}
Summary
- The universal gas law is \begin{align*}PV = nRT\end{align*}
PV=nRT , where \begin{align*}P\end{align*}P is pressure, \begin{align*}V\end{align*}V is volume, \begin{align*}n\end{align*}n is number of moles, \begin{align*}R\end{align*}R is the universal gas law constant, and \begin{align*}T\end{align*}T is the absolute temperature. - The value of \begin{align*}R\end{align*}
R varies depending on the units used for \begin{align*}P\end{align*}P and \begin{align*}V\end{align*}V . Two common values are \begin{align*}0.0821 \ \text{L} \cdot \text{atm/mol} \cdot \text{K}\end{align*}0.0821 L⋅atm/mol⋅K and \begin{align*}R = 8.314 \ \text{J/mol} \cdot \text{K}\end{align*}R=8.314 J/mol⋅K . - STP is standard temperature and pressure; 273 K and 1.00 atm.
- One mole of a gas at STP has a volume of 22.4 liters.
Practice
The following video discusses the constant \begin{align*}R\end{align*}
- Why is it important to have values for R in kPa, atm, and mmHg?
- Why do the units of R include pressure, temperature, volume, and moles?
Instruction and practice problems involving the universal gas law:
http://www.sparknotes.com/testprep/books/sat2/chemistry/chapter5section10.rhtml
Review
- The initial pressure in a helium gas cylinder is 30 atm. After many balloons have been blown up, the pressure in the cylinder has decreased to 6 atm while the volume and temperature remain the same. What fraction of the original amount of gas remains in the cylinder?
- Calculate the volume of 8.88 mol of helium gas at 20.0°C and 1.19 atm pressure.
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Image Attributions
- State the universal gas law.
- State the universal gas law constant, R.
- Given three of the four unknowns in the universal gas law, solve for the fourth.
Concept Nodes:
- Universal gas law: @$\begin{align*}PV = nRT \end{align*}@$
- Universal gas law constant: Two common values are @$\begin{align*}0.0821 \ L \cdot atm/mol \cdot K\end{align*}@$ and @$\begin{align*}R = 8.314 \ J/mol \cdot K\end{align*}@$.