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Advanced Physics Topics like quantum mechanics and relativity have revolutionized our understanding of the universe.
What is the Pressure Law in Simple Terms?
Charle’s Law
The relationship between the volume and temperature of a fixed mass of gas at constant pressure was investigated experimentally by the French scientist Charles in 1787. He concluded that at constant pressure a fixed volume of all gases would expand by the same amount for an equal rise in temperature. Gay Lussac arrived at almost the same result in 1802. He found that the coefficient of volume expansion of all gases has the same value if pressure is kept constant. In 1842 Regnault showed that the experimental value of this coefficient is \(\frac{1}{273}\) per degree Celsius. Charles’ law is enunciated combining the experimental results of Gay Lussac and Regnault.
Statement: When pressure is kept constant, volume of a fixed mass of gas increases or decreases by \(\frac{1}{273}\) th part of its volume at 0°C, for each degree Celsius rise or fall in temperature.
Mathematical expression: Let V0 be the volume of a mass m of a gas at 0°C. As per Charles’ law, the volume at 1°C will be
Similarly, if temperature is decreased by t°C, i.e., at a temperature -t° C, the volume of the gas becomes
V-t = V0(1 – \(\frac{t}{273}\))
Therefore, at constant pressure volume of a fixed mass of gas changes linearly with its temperature. So, at fixed pressure, a graph plotted between the volume of a gas of a fixed mass and its temperature, gives a straight line (AB) [Fig].
Pressure Law
The law that relates change in pressure of a fixed mass of a gas at fixed volume, with change in temperature is called pressure law or Regnault’s law.
Statement: When volume is kept constant, pressure of a fixed mass of gas increases or decreases by \(\frac{1}{273}\) th part of its pressure at 0°C, for each degree centigrade rise or fall in temperature.
Mathematical expression: Let p0 be the pressure of a fixed mass of a gas at 0°C. The pressure is pt when temperature is raised to t °C .
Therefore the pressure of the gas at 1°C,
P1 = P0 + \(\frac{p_0}{273}\) = p0(1 + \(\frac{1}{273}\)) and the pressure of the gas at t °C,
pt = p0 + \(\frac{p_0 t}{273}\) = p0(1 + \(\frac{t}{273}\)) ……. (1)
Similarly, the pressure of the gas at -t °C ,
p-t = po – \(\frac{p_0 t}{273}\) = po(1 – \(\frac{t}{273}\))
Hence, the change in pressure of a fixed mass of gas is linearly related to the change in temperature at constant volume.
So, at constant volume, a graph plotted between the pressure of a gas of fixed mass and its temperature gives a straight line (AB) [Fig.],
The increase in either the volume, or the pressure of a gas with a rise in temperature is loosely termed as thermal expansion, although an increase in pressure is not actually an expansion.
Alternative Forms of Charles’ Law And Pressure Law
Charle’s Law:
Suppose a fixed mass of a gas, at constant pressure, has volume V0 at 0°C, V1 at t1°C and V2 at t2°C. From Charles’ law,
V1 =V0(1 + \(\frac{t_1}{273}\)) = V0\(\left(\frac{273+t_1}{273}\right)\) = \(\frac{V_0}{273}\) ᐧ T1
[where T1 = t1 + 273]
Obviously, the temperature t1 in Celsius scale is the same as the temperature T1 K in Kelvin scale.
Similarly, V2 = \(\frac{V_0}{273}\) ᐧ T2 [where T2 = t2 + 273]
Here, t2° C = T2K.
∴ \(\frac{V_1}{V_2}\) = \(\frac{T_1}{T_2}\) = constant
or, V ∝ T at constant pressure.
Hence, Charles’ law can also be stated as, the volume of a fixed mass of gas at constant pressure is directly proportional to its temperature in absolute scale.
Pressure law: Suppose a fixed mass of a gas at constant volume has pressure p0 at 0°C, p1 at t1°C and p2 at t2°C. Hence, from pressure law,
p1 = p0(1 + \(\frac{t_1}{273}\)) = \(\frac{p_0}{273}\)(273 + t1) = \(\frac{p_0}{273}\) × T1
[where T1 = t1 + 273]
It is clear that the temperature t °C is the same as the temperature T1 K.
Similarly, p2 = \(\frac{p_0}{273}\) × T2 [where T2 = t2 + 273]
Here, t2°C = T2K.
∴ \(\frac{p_1}{p_2}\) = \(\frac{T_1}{T_2}\) or, p ∝ T, at constant volume.
Hence, pressure law can be expressed as, the pressure of a fixed mass of a gas at constant volume is directly proportional to its temperature in absolute scale.
Combination of Boyle’s Law And Charles Law: Equation of State of an Ideal Gas
Ideal or Perfect gas: The gases which obey Boyle’s and Charles’ law at any temperature are called ideal gases.
The equation obtained by combining Boyle’s law and Charles’ law is called the equation of state of an ideal gas.
From Boyle’s law, V ∝ \(\frac{1}{p}\) when T is constant.
From Charles’ law, V ∝ T when p is constant.
∴ V ∝ \(\frac{T}{p}\) when both p and T vary
or, pV = kT ……. (1)
where k is a constant whose value depends on the units of p, V, T and the mass of the gas. If a gas of fixed mass occupies a volume V<sub>1</sub> at pressure p<sub>1</sub> and temperature T<sub>1</sub>, and after a general change, a volume V<sub>2</sub> at pressure p<sub>2</sub> and temperature T<sub>2</sub>, then
\(\frac{p_1 V_1}{T_1}\) = \(\frac{p_2 V_2}{T_2}\) ….. (2)
It is known that the physical property of an ideal gas of fixed mass depends on its pressure, volume and temperature.
Hence, the equation pV = kT is called the equation of state of an ideal gas.
Note that no real gas totally follows this equation of state. The deviation, though small at ordinary temperature and pressure, becomes significant at high pressure and at low temperature.