Question

State (i) Van’t Hoff-Boyles law (ii) Van’t Hoff Charles law

Answer 1

Hint: Both the laws given in question are derived and stated from the ideal gas equation. Boyle’s law states that pressure is inversely proportional to the volume you can imagine this law by taking a piston as an example. Charle’s law states that volume is directly proportional to temperature. Imagine an example of boiling water.

Complete step-by-step answer:
(i) Van’t Hoff-Boyle;s law:
Boyle's law states that at constant temperature the pressure exerted by a fixed mass of a gas is inversely proportional to volume occupied by the gas.
$P\alpha \dfrac{1}{V}$,at constant temperature
PV=constant
Where,
P= pressure of a enclosed gas
V=volume of a enclosed gas
Boyle's law can be derived on the basis of kinetic theory of gases.
 Consider a gas of volume V and N molecules, each of mass m.
Mass of gas M=Nm
According to kinetic theory of gases pressure exerted is given by
$\begin{align}
  & P=\dfrac{1}{3}\rho C_{RMS}^{2} \\
 & P=\dfrac{1}{3}\dfrac{M}{V}C_{RMS}^{2} \\
 & PV=\dfrac{1}{3}\dfrac{M}{1}C_{RMS}^{2} \\
 & PV=\dfrac{2}{3}N\left( \dfrac{1}{2}mC_{RMS}^{2} \right) \\
\end{align}$
The quantity $\dfrac{1}{2}mC_{RMS}^{2}$. The average kinetic energy of gas molecules remains constant at constant temperature according to the assumption.
Therefore, PV=Constant
$P\alpha \dfrac{1}{V}$, and this is Boyle’s law.

(ii)Van’t Hoff Charle’s:
Charle’s law states that at a fixed pressure the volume is proportional to its absolute temperature.
I.e. $P\alpha T$
$\dfrac{V}{T}=k=cons\tan t$
Where k is the constant which depends on pressure of gas, the amount of gas and also the unit of volume.
If ${{V}_{1}} and {{T}_{1}}$ are the initial values of volume and temperature of a gas then, $\dfrac{{{V}_{1}}}{{{T}_{1}}}=k$
Also, if the temperature is now changed to ${{T}_{2}}$ such that the volume changes to ${{V}_{2}}$
It can be written as
$\dfrac{{{V}_{2}}}{{{T}_{2}}}=k$
Therefore we can write as,
\[\begin{align}
  & \dfrac{{{V}_{1}}}{{{T}_{1}}}=\dfrac{{{V}_{2}}}{{{T}_{2}}} \\
 & {{V}_{1}}{{T}_{1}}={{V}_{2}}{{T}_{2}} \\
\end{align}\]
This is Charle’s law.

Note: Constant in Boyle’s law depends on pressure and volume of gas contained in the container. While constant in charles's law depends on temperature and volume. Students should not be mug derivation and concept. You can get all relations from the ideal gas equation.