How can we use the equation PV = NkT? At first, it may seem that not enough information is given, because the volume V and number of atoms N are not specified. We know the initial pressure P 0 = 7.00 × 10 5 Pa, the initial temperature T 0 = 18.0✬, and the final temperature T f = 35.0✬. First we need to identify what we know and what we want to know, and then identify an equation to solve for the unknown. ![]() The pressure in the tire is changing only because of changes in temperature. What is the pressure after its temperature has risen to 35.0✬? Assume that there are no appreciable leaks or changes in volume. Suppose your bicycle tire is fully inflated, with an absolute pressure of 7.00 × 10 5 Pa (a gauge pressure of just under 90.0 lb/in 2) at a temperature of 18.0✬. Calculating Pressure Changes Due to Temperature Changes: Tire Pressure Once the volume of the tire is constant, the equation PV = NkT predicts that the pressure should increase in proportion to the number N of atoms and molecules.Įxample 1. At first, the pressure P is essentially equal to atmospheric pressure, and the volume V increases in direct proportion to the number of atoms and molecules N put into the tire. Let us see how the ideal gas law is consistent with the behavior of filling the tire when it is pumped slowly and the temperature is constant. (Note, for example, that N is the total number of atoms and molecules, independent of the type of gas.) The ideal gas law describes the behavior of real gases under most conditions. In the ideal gas model, the volume occupied by its atoms and molecules is a negligible fraction of V. The ideal gas law can be derived from basic principles, but was originally deduced from experimental measurements of Charles’ law (that volume occupied by a gas is proportional to temperature at a fixed pressure) and from Boyle’s law (that for a fixed temperature, the product PV is a constant). The constant k is called the Boltzmann constant in honor of Austrian physicist Ludwig Boltzmann (1844–1906) and has the value k = 1.38 × 10 −23 J/K. The ideal gas law states that PV = NkT, where P is the absolute pressure of a gas, V is the volume it occupies, N is the number of atoms and molecules in the gas, and T is its absolute temperature. In contrast, in liquids and solids, atoms and molecules are closer together and are quite sensitive to the forces between them. The motion of atoms and molecules (at temperatures well above the boiling temperature) is fast, such that the gas occupies all of the accessible volume and the expansion of gases is rapid. Because atoms and molecules have large separations, forces between them can be ignored, except when they collide with each other during collisions. The answer lies in the large separation of atoms and molecules in gases, compared to their sizes, as illustrated in Figure 2. This raises the question as to why gases should all act in nearly the same way, when liquids and solids have widely varying expansion rates. In addition, you will note that most gases expand at the same rate, or have the same β. The large coefficients mean that gases expand and contract very rapidly with temperature changes. We can see evidence of this in Table 1 in Thermal Expansion of Solids and Liquids, where you will note that gases have the largest coefficients of volume expansion. We will primarily use the term “molecule” in discussing a gas because the term can also be applied to monatomic gases, such as helium.) (Most gases, for example nitrogen, N 2, and oxygen, O 2, are composed of two or more atoms. In particular, we examine the characteristics of atoms and molecules that compose gases. In this section, we continue to explore the thermal behavior of gases. As a result, the balloon experiences a buoyant force pushing it upward. The air inside this hot air balloon flying over Putrajaya, Malaysia, is hotter than the ambient air.
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