Contents
The laws of Physics Topics are used to explain everything from the smallest subatomic particles to the largest galaxies.
What are the Parameters that Determine the Physical State of a Gas?
Introduction
Gases expand on heating and contract on cooling, like solids and liquids. Like liquids, gases too do not have any fixed shape and therefore the linear or the surface expansion of a gas is irrelevant. Only the change in volume with the change in temperature is of importance. Other characteristic features of gaseous expansions, compared to those of solids and liquids, are discussed below.
i) Coefficient of volume expansion for a gas has a much higher value than those for solids and liquids. For a certain rise in temperature, expansion of the container is negligible compared to the expansion of the gas in it. Hence, apparent expansion of gas is practically the same as real expansion, and usually not reckoned separately unless a very high accuracy is required.
ii) Unlike solids and liquids, volume of a fixed mass of a gas is considerably affected due to any change in pressure. So, the effects of both temperature and pressure have to be studied in connection with the expansion or contraction of a gas. While studying the effect of one, the other is usually kept constant.
iii) The rate of expansion or contraction, due to the change in either pressure or temperature, does not differ for different gases. Unlike solid and liquids, the coefficient of expansion is the same for all gases.
The state of a fixed mass of gas is therefore determined by the parameters
- volume,
- pressure and
- temperature.
The rules, that govern the change of one parameter with the change of another keeping the third one constant, are called gas laws.
1. It should be mentioned here that the pressure of a gas means the pressure exerted by the gas. In equilibrium (i.e., when each parameter of the gas is a constant) the pressure exerted by the gas and the pressure applied on the gas are equal.
2. The gas which follows Boyle’s law and Chajles’ law at any temperature and pressure is called an ideal gas. In reality, an ideal gas does not exist. However, the ideal gas concept provides a very useful tool for the analysis of real gases.
Boyle’s Law
The relation between the volume and the pressure of a gas, at constant temperature, was first proposed by the British scientist Robert Boyle in 1660 AD, and that relation is called Boyle’s law.
Statement: At constant temperature, volume of a fibred mass of gas is inversely proportional to its pressure.
Mathematically, if V is the volume of a gas of certain mass, and p is its pressure, then according to Boyle’s law,
V ∝ \(\frac{1}{p}\), when temperature remains constant
or V = \(\frac{k}{p}\) or, pV = k = constant
The value of k depends on the mass and the temperature of the gas. Hence, the product of the pressure and the volume of a fixed mass of gas at a fixed temperature remains constant.
Thus at constant temperature, if V1, V2, V3 are volumes of a fixed mass of a gas at pressures p1, p2, p3…….., then according to this law,
p1V1 = p2V2 = p3V3….. = k (a constant).
Graphical representation of Boyle’s law :Boyle’s law can be expressed through different graphs:
1. p-V graph: Plottingvolumes Vofafixed mass of gas, at constant temperature, along the horizontal axis and the corresponding pressures along temperature the vertical axis, the obtained graph is a rectangular hyperbola [Fig.]. Such a graph is called the Isothermal of the gas. At different values of fixed temperatures, the isothermals are different rectangular hyperbolas.
2. \(\frac{1}{V}\)-p graph: For a fixed mass of a gas at constant temperature, the graph of pressure p plotted along the horizontal axis and reciprocal \(\frac{1}{V}\) of volume V plotted along the vertical axis, is a straight
line passing through the origin [Fig]. At different values of fixed temperatures, \(\frac{1}{V}\)-p graphs will be different straight lines.
A V-\(\frac{1}{p}\) graph is also of the same graph.
3. pV-p graph:Graph obtained, by plotting pressures p of a fixed mass of gas at constant temperature along the horizontal axis and the corresponding products pV along the vertical axis, is a straight line parallel to the p axis [Fig.]. At different fixed temperatures, pV-p graphs will be different parallel lines.
At constant temperature, pV-V graph is also of the same nature.
These graphs, obtained experimentally, verify Boyle’s law.
Numerical Examples
Example 1.
The volume of a fixed mass of gas at STP is 500 cm3. What will be its volume at 700 mmHg pressure if its temperature remains constant?
Solution:
As temperature remains constant, Boyle’s law is
applicable.
Given, p1 = 76 cmHg, V1 = 500 cm3, p2 = 70 cmHg.
∴ p1V1 = P2V2 ∴ 76 × 500 = 70 × V2
or, V2 = \(\frac{76 \times 500}{70}\) = 542.86 cm3.
Example 2.
While tabulating the pressures and volumes for a fixed mass of a gas at a fixed temperature, a student forgets to record a few observations, as shown below. Fill in the blanks.
Solution:
Since the temperature is fixed, Boyle’s law is applicable here, i.e., pV = constant.
From reading 1 and 3, p1V1 = 1oo × 80 = 8000 and p3V3 = 200 × 40 = 8000. Hence the value of the constant in this case is 8000.
∴ For reading (2),
125 × V2 = 8000 or, V2 = \(\frac{8000}{125}\) = 64 cm3
and for reading (4)
p4 × 32 = 8000 or, p4 = \(\frac{8000}{32}\) = 250 mmHg.
Example 3.
The volume of a gas at 1 standard atmosphere is compressed to \(\frac{1}{6}\)th of its value at constant temperature. What will be its final pressure?
Solution:
Let the initial volume at 1 standard atmosphere be x cm3 and the final pressure be p.
The temperature remains constant; so applying Boyle’s law, we get,
1 × x = p × \(\frac{x}{6}\) or, p = 6 standard atmospheres.
Example 4.
A 100 cm long vertical cylinder, closed at the bottom end, has a movable, frictionless, air tight disc attached at its other end. An Ideal gas is confined within the cylinder. Initially when the disc between the confined gas and atmosphere is in equilibrium, the length of the gas column is 90 cm. Mercury is poured slowly on the disc. When the disc descends by 32 cm, mercury over It is just about to overflow. Find the atmospheric pressure if the operation took place at a constant temperature of the gas. Neglect the weight or thickness of the disc.
Solution:
Let the atmospheric pressure be p cm Hg and α = area of cross-section of the cylinder. Initially in equilibrium, pressure of confined gas = atmospheric pressure p.
Now, volume of enclosed gas V = 90α cm3.
When mercury just overflows from the cylinder, pressure of the gas p1 = (p + 42) cmHg [Fig.] and volume V1 = (90 – 32)α = 58αcm3.
As the temperature is constant, according to Boyle’s law,
pV = p1V1
or, p × 90α = (p + 42) × 58α
90p = 58p + 58 × 42
or, 32p = 58 × 42
or, p = \(\frac{58 \times 42}{32}\) = 76.125 cmHg