Question

State Boyle's law.

Answer 1

Hint: There are three primary gas laws. Boyle's law is one of them. Boyle’s law is derived from the ideal gas equation. It is known as pressure-volume relationship. Because, it states the relation between pressure P of a given mass and volume V when a constant temperature T and mass of the gas is fixed.

Complete answer:
Boyle’s law states that at constant temperature, the pressure exerted by a gas is inversely proportional to the volume occupied by it. In other words we can say, volume and pressure are inversely proportional to each other but only at constant temperature and constant mass of gas.
For a gas, the relation between volume and pressure can be expressed as,
$P\propto \dfrac { 1 }{ V }$ …(1)
Where, P: Pressure exerted by the gas
             V: Volume occupied by the gas
Writing the above proportionality in equation form we get,
$P=\dfrac { k }{ V }$
Where, k is a constant
According to Kinetic theory of gases, pressure exerted is given by,
$P=\dfrac { 1 }{ 3 } \rho { v }_{ RMS }^{ 2 }$...(2)
Where, ${ v }_{ RMS }$: Root mean square velocity
But, $\rho =\dfrac { M }{ V }$
Where, M: Total Mass of gas= Nm
Therefore, equation. (1) becomes,
$P=\dfrac { M }{ 3V } { v }_{ RMS }$
$\therefore PV=\dfrac { M }{ 3 } { v }_{ RMS }^{ 2 }$
Now, multiplying and dividing by 2 we get,
$\therefore PV=\dfrac { 2M }{ 3 } { \dfrac { 1 }{ 2 } v }_{ RMS }^{ 2 }$
Substituting M by nm we get,
$ PV=\dfrac { 2N }{ 3 } { \dfrac { 1 }{ 2 } mv }_{ RMS }^{ 2 }$
We know, $K.E.={ \cfrac { 1 }{ 2 } mv }_{ RMS }^{ 2 }$
Therefore, average kinetic energy remains constant and we get PV= constant.
Thus, according to Boyle’s law, $P\propto \dfrac { 1 }{ V }$ .

Note:
The constant k depends upon the pressure and volume of the gas contained in the container. This law can be derived from the ideal gas equation. Boyle’s law, Charles’s law , Gay-Lussac's law and Avogadro's hypothesis together gives ideal gas law. Bicycle pumps and working of syringes are few real-life applications of Boyle's law.