Question

The tyre of a car contains air at $15 ^\circ C$ .If the temperature increases to $25 ^\circ C$ .The approximate percentage rise in pressure is
A. $7$
B. $9$
C. $11$
D. $13$

Answer 1

There are various parameters which govern the state of particles which exist in gaseous state.Some of these parameters include Pressure,Temperature,volume etc.Over the time laws have found out which determine the inter-dependence of these parameters.

Complete answer:
In the above given question we are considering two of gaseous state parameters,they are temperature and pressure.Since we need to establish a relationship between these two we need to consider a gas law which will relate these two parameters.The relationship between temperature and pressure is given by Charles Law which states that,” Charles law states that the volume of an ideal gas is directly proportional to the absolute temperature at constant pressure”.
The mathematical expression for which is given by,” $P\alpha T$ “
In the above question,by converting temperature from Celsius to Kelvin
$K = 275 + C$ ,where K represents temperature in Kelvin and C represents temperature in Celsius.
Converting the temperature from Celsius to Kelvin
$T_1$ = 290K
$T_2$ = 310K
Let P_1 be the pressure in the tyre when temperature is at $290K$ and P_2 be the pressure in the tyre when temperature is at $310K$ .
$\dfrac{{P_2}}{{P_1}} = \dfrac{{T_2}}{{T_1}}$ (Using Charles's law)
By implementing the property of dividendo
$\dfrac{{(P_2 - P_1)}}{{P_1}} \times 100 = \dfrac{{(T_2 - T_1)}}{{T_1}} \times 100$
Since we know that the values of difference in temperature we will substitute them,
$\dfrac{{\vartriangle P}}{{P_1}} \times 100 = \dfrac{{20}}{{298}} \times 100$
Further simplifying the equation,
$\dfrac{{\vartriangle P}}{{P_1}} \times 100 = 0.0698 \times 100$
Finally solving the equation to get the correct values
$\dfrac{{\vartriangle P}}{{P_1}} \cong 7\%$
Therefore tallying our answer with the given options.

Option (a) is correct.

Note:
The ideal gas law which is given by $PV = nRT$ has formed as the base of all the gas laws.It gives a good approximation of the behaviour of various gases under ideal conditions.However it has its own set of limitations.