Answer :
(a) The pressure inside the bottle just after it filled is 19.8 atm.
(b) The pressure inside the bottle as it opened in the 21.0°C is 11.8 atm.
Explanation :
<u>Part (a) :</u>
Boyle's Law : It is defined as the pressure of the gas is inversely proportional to the volume of the gas at constant temperature and number of moles.
![P\propto \frac{1}{V}](https://tex.z-dn.net/?f=P%5Cpropto%20%5Cfrac%7B1%7D%7BV%7D)
or,
![P_1V_1=P_2V_2](https://tex.z-dn.net/?f=P_1V_1%3DP_2V_2)
where,
= initial pressure of gas = 1.01 atm
= final pressure of gas = ?
= initial volume of gas = 20.6 L
= final volume of gas = 1.05 L
Now put all the given values in the above equation, we get:
![1.01atm\times 20.6L=P_2\times 1.05L](https://tex.z-dn.net/?f=1.01atm%5Ctimes%2020.6L%3DP_2%5Ctimes%201.05L)
![P_2=19.8atm](https://tex.z-dn.net/?f=P_2%3D19.8atm)
Therefore, the pressure inside the bottle just after it filled is 19.8 atm.
<u>Part (b) :</u>
Gay-Lussac's Law : It is defined as the pressure of the gas is directly proportional to the temperature of the gas at constant volume and number of moles.
![P\propto T](https://tex.z-dn.net/?f=P%5Cpropto%20T)
or,
![\frac{P_1}{T_1}=\frac{P_2}{T_2}](https://tex.z-dn.net/?f=%5Cfrac%7BP_1%7D%7BT_1%7D%3D%5Cfrac%7BP_2%7D%7BT_2%7D)
where,
= initial pressure of gas = 19.8 atm
= final pressure of gas = ?
= initial temperature of gas = ![220.0^oC=273+220.0=493.0K](https://tex.z-dn.net/?f=220.0%5EoC%3D273%2B220.0%3D493.0K)
= final temperature of gas = ![21.0^oC=273+21.0=294.0K](https://tex.z-dn.net/?f=21.0%5EoC%3D273%2B21.0%3D294.0K)
Now put all the given values in the above equation, we get:
![\frac{19.8atm}{493.0K}=\frac{P_2}{294.0K}](https://tex.z-dn.net/?f=%5Cfrac%7B19.8atm%7D%7B493.0K%7D%3D%5Cfrac%7BP_2%7D%7B294.0K%7D)
![P_2=11.8atm](https://tex.z-dn.net/?f=P_2%3D11.8atm)
Therefore, the pressure inside the bottle as it opened in the 21.0°C is 11.8 atm.