Answer :
(a). The final temperature of the gas in the cylinder A is 320 K.
(b). The final temperature of the gas in the cylinder B is 233.7 K.
(c). The final volume of the gas in the cylinder A is ![7.86\times10^{-3}\ m^3](https://tex.z-dn.net/?f=7.86%5Ctimes10%5E%7B-3%7D%5C%20m%5E3)
(d). The final volume of the gas in the cylinder B is ![5.7\times10^{-3}\ m^3](https://tex.z-dn.net/?f=5.7%5Ctimes10%5E%7B-3%7D%5C%20m%5E3)
Explanation :
Given that,
Number of mole n = 0.30 mol
Initial temperature = 320 K
Pressure = 3.0 atm
Final pressure = 1.0 atm
We need to calculate the initial volume
Using formula of ideal gas
![P_{1}V_{1}=nRT](https://tex.z-dn.net/?f=P_%7B1%7DV_%7B1%7D%3DnRT)
![V_{1}=\dfrac{nRT}{P_{1}}](https://tex.z-dn.net/?f=V_%7B1%7D%3D%5Cdfrac%7BnRT%7D%7BP_%7B1%7D%7D)
Put the value into the formula
![V_{1}=\dfrac{0.30\times8.314\times320}{3.039\times10^{5}}](https://tex.z-dn.net/?f=V_%7B1%7D%3D%5Cdfrac%7B0.30%5Ctimes8.314%5Ctimes320%7D%7B3.039%5Ctimes10%5E%7B5%7D%7D)
![V_{1}=2.62\times10^{-3}\ m^3](https://tex.z-dn.net/?f=V_%7B1%7D%3D2.62%5Ctimes10%5E%7B-3%7D%5C%20m%5E3)
(a). We need to calculate the final temperature of the gas in the cylinder A
Using formula of ideal gas
In isothermally, the temperature is not change.
So, the final temperature of the gas in the cylinder A is 320 K.
(b). We need to calculate the final temperature of the gas in the cylinder B
Using formula of ideal gas
![T_{2}=T_{1}\times(\dfrac{P_{1}}{P_{2}})^{\frac{1}{\gamma}-1}](https://tex.z-dn.net/?f=T_%7B2%7D%3DT_%7B1%7D%5Ctimes%28%5Cdfrac%7BP_%7B1%7D%7D%7BP_%7B2%7D%7D%29%5E%7B%5Cfrac%7B1%7D%7B%5Cgamma%7D-1%7D)
Put the value into the formula
![T_{2}=320\times(\dfrac{3}{1})^{\frac{1}{1.4}-1}](https://tex.z-dn.net/?f=T_%7B2%7D%3D320%5Ctimes%28%5Cdfrac%7B3%7D%7B1%7D%29%5E%7B%5Cfrac%7B1%7D%7B1.4%7D-1%7D)
![T_{2}=233.7\ K](https://tex.z-dn.net/?f=T_%7B2%7D%3D233.7%5C%20K)
(c). We need to calculate the final volume of the gas in the cylinder A
Using formula of volume of the gas
![P_{1}V_{1}=P_{2}V_{2}](https://tex.z-dn.net/?f=P_%7B1%7DV_%7B1%7D%3DP_%7B2%7DV_%7B2%7D)
![V_{2}=\dfrac{P_{1}V_{1}}{P_{2}}](https://tex.z-dn.net/?f=V_%7B2%7D%3D%5Cdfrac%7BP_%7B1%7DV_%7B1%7D%7D%7BP_%7B2%7D%7D)
Put the value into the formula
![V_{2}=\dfrac{3\times2.62\times10^{-3}}{1}](https://tex.z-dn.net/?f=V_%7B2%7D%3D%5Cdfrac%7B3%5Ctimes2.62%5Ctimes10%5E%7B-3%7D%7D%7B1%7D)
![V_{2}=0.00786\ m^3](https://tex.z-dn.net/?f=V_%7B2%7D%3D0.00786%5C%20m%5E3)
![V_{2}=7.86\times10^{-3}\ m^3](https://tex.z-dn.net/?f=V_%7B2%7D%3D7.86%5Ctimes10%5E%7B-3%7D%5C%20m%5E3)
(d). We need to calculate the final volume of the gas in the cylinder B
Using formula of volume of the gas
![V_{2}=V_{1}(\dfrac{P_{1}}{P_{2}})^{\frac{1}{\gamma}}](https://tex.z-dn.net/?f=V_%7B2%7D%3DV_%7B1%7D%28%5Cdfrac%7BP_%7B1%7D%7D%7BP_%7B2%7D%7D%29%5E%7B%5Cfrac%7B1%7D%7B%5Cgamma%7D%7D)
![V_{2}=2.62\times10^{-3}\times(\dfrac{3}{1})^{\frac{1}{1.4}}](https://tex.z-dn.net/?f=V_%7B2%7D%3D2.62%5Ctimes10%5E%7B-3%7D%5Ctimes%28%5Cdfrac%7B3%7D%7B1%7D%29%5E%7B%5Cfrac%7B1%7D%7B1.4%7D%7D)
![V_{2}=0.0057\ m^3](https://tex.z-dn.net/?f=V_%7B2%7D%3D0.0057%5C%20m%5E3)
![V_{2}=5.7\times10^{-3}\ m^3](https://tex.z-dn.net/?f=V_%7B2%7D%3D5.7%5Ctimes10%5E%7B-3%7D%5C%20m%5E3)
Hence, (a). The final temperature of the gas in the cylinder A is 320 K.
(b). The final temperature of the gas in the cylinder B is 233.7 K.
(c). The final volume of the gas in the cylinder A is ![7.86\times10^{-3}\ m^3](https://tex.z-dn.net/?f=7.86%5Ctimes10%5E%7B-3%7D%5C%20m%5E3)
(d). The final volume of the gas in the cylinder B is ![5.7\times10^{-3}\ m^3](https://tex.z-dn.net/?f=5.7%5Ctimes10%5E%7B-3%7D%5C%20m%5E3)