Question

How can Gay Lussac’s law can be derived from the combined gas law?

Answer 1

Hint: The combined gas law is one which tends to combine the Charles law, Gay Lussac law and the Boyles law. These laws tend to relate one thermodynamic variable to the other and thus hold everything constant. The formula of this law includes the pressure, temperature and volume.

Complete step by step answer:
The Gay Lussac law says that the pressure which has been exerted by the gas tends to vary directly with the absolute temperature of the gas. In this law the pressure of the substance is directly proportional to the temperature. The mathematical expression to represent this law is:
$$P \propto T$$ So,
$$\dfrac{P}{T} = k$$
P is the pressure which has been exerted by the gas
T represent the temperature of the gas
The k is the constant here.
So to derive the Gay Lussac law from combined gas law we need to start from the ideal gas law. As the combined gas law has been derived from ideal gas law. so the ideal gas law is:
$$PV = nRT$$
Defining the initial and final state of the substance with the constant V and n.$$({V_1} = {V_2} \equiv V,{n_1} = {n_2} \equiv n)$$
So according to this we can write the ideal gas law for two substances as the following:
$${P_1}V = nR{T_1}$$ ……1
$${P_2}V = nR{T_2}$$……2
So dividing equation 2 by equation 1 we get,
$$\dfrac{{{P_2}V}}{{{P_1}V}} = \dfrac{{nR{T_2}}}{{nR{T_1}}}$$
On simplifying we get,
$$\dfrac{{{P_2}}}{{{P_1}}} = \dfrac{{{T_2}}}{{{T_1}}}$$
On rearranging the variables we get,
$$\dfrac{{{P_2}}}{{{T_2}}} = \dfrac{{{P_1}}}{{{T_1}}}$$
This above relation is the Gay Lussac law.

Note: The ideal gases are those which have elastic collisions between the molecules and also do not have intermolecular attractive forces. The ideal gas law includes the Charles law, Boyle's law and Avogadro law. the ideal gases obey this law. The ideal gases have elastic collisions and temperature of the gas is directly proportional to the average kinetic energy of the molecules.