Question

The relation between volume V, pressure P and absolute temperature T of an ideal gas is PV=XT, where X is a constant. The value of X depends upon
A. the mass of gas molecule
B. the average kinetic energy of the gas molecules
C. p, V and T
D. The number of gas molecules in V

Answer 1

Hint First, we will derive the ideal gas equation using the different laws such as Boyle’s law, Charles’s law and Avogadro’s law. Then we will equate the both equations and find the value of x.

Complete step-by-step solution The ideal gas law is the equation of state of ideal gas.
Derivation of ideal gas equation
Let pressure exerted by the gas=P, Temperature=T, Volume of the gas=V, moles=n, universal gas constant=R
Acc. To Boyle’s law,
It states that volume is inversely proportional to pressure given temperature remains the same.
 \[V \propto \dfrac{1}{P}\]
Acc. To Charle’s law,
It states that volume of the gas occupied is directly proportional to temperature given pressure is constant.
 \[V \propto T\]
Acc. To Avogadro’s law ,
 \[V \propto n\]
Combining all three equations
 \[V \propto \dfrac{{nT}}{P}\]
Ideal gas equation,
 \[PV = nRT\] , R= Universal Gas Constant=8.314J/mol-K ……(1)
We are given that
 \[PV = xT\] …….(2)
Comparing the two equations
 \[x = nR\]
 \[n = \dfrac{N}{{{N_a}}}\] , N= number of molecules of the gas
                   \[{N_a}\] = Avogadro’s number
So \[{N_a}\] and R are constant
x depends upon N= number of molecules of gas

option(d) the number of gas molecules in V

Note
1. No gas is the ideal gas, it is a good approximation of the behaviour of many gases under various conditions but these are under several limitations.
2. The ideal gas model depends on some assumptions such as molecules of gas are small spheres, indistinguishable. All collisions are elastic.