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The ideal gas law can be considered to be another manifestation of the law of conservation of energy (see Conservation of Energy ). Work done on a gas results in an increase in its energy, increasing pressure and/or temperature, or decreasing volume. This increased energy can also be viewed as increased internal kinetic energy, given the gas’s atoms and molecules.
Let us now examine the role of energy in the behavior of gases. When you inflate a bike tire by hand, you do work by repeatedly exerting a force through a distance. This energy goes into increasing the pressure of air inside the tire and increasing the temperature of the pump and the air.
The ideal gas law is closely related to energy: the units on both sides are joules. The right-hand side of the ideal gas law in $\text{PV}=\text{NkT}$ is $\text{NkT}$ . This term is roughly the amount of translational kinetic energy of $N$ atoms or molecules at an absolute temperature $T$ , as we shall see formally in Kinetic Theory: Atomic and Molecular Explanation of Pressure and Temperature . The left-hand side of the ideal gas law is $\text{PV}$ , which also has the units of joules. We know from our study of fluids that pressure is one type of potential energy per unit volume, so pressure multiplied by volume is energy. The important point is that there is energy in a gas related to both its pressure and its volume. The energy can be changed when the gas is doing work as it expands—something we explore in Heat and Heat Transfer Methods —similar to what occurs in gasoline or steam engines and turbines.
Step 1 Examine the situation to determine that an ideal gas is involved. Most gases are nearly ideal.
Step 2 Make a list of what quantities are given, or can be inferred from the problem as stated (identify the known quantities). Convert known values into proper SI units (K for temperature, Pa for pressure, ${\text{m}}^{3}$ for volume, molecules for $N$ , and moles for $n$ ).
Step 3 Identify exactly what needs to be determined in the problem (identify the unknown quantities). A written list is useful.
Step 4 Determine whether the number of molecules or the number of moles is known, in order to decide which form of the ideal gas law to use. The first form is $\text{PV}=\text{NkT}$ and involves $N$ , the number of atoms or molecules. The second form is $\text{PV}=\text{nRT}$ and involves $n$ , the number of moles.
Step 5 Solve the ideal gas law for the quantity to be determined (the unknown quantity). You may need to take a ratio of final states to initial states to eliminate the unknown quantities that are kept fixed.
Step 6 Substitute the known quantities, along with their units, into the appropriate equation, and obtain numerical solutions complete with units. Be certain to use absolute temperature and absolute pressure.
Step 7 Check the answer to see if it is reasonable: Does it make sense?
Liquids and solids have densities about 1000 times greater than gases. Explain how this implies that the distances between atoms and molecules in gases are about 10 times greater than the size of their atoms and molecules.
Atoms and molecules are close together in solids and liquids. In gases they are separated by empty space. Thus gases have lower densities than liquids and solids. Density is mass per unit volume, and volume is related to the size of a body (such as a sphere) cubed. So if the distance between atoms and molecules increases by a factor of 10, then the volume occupied increases by a factor of 1000, and the density decreases by a factor of 1000.
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