0.0We are asked to determine the volume of a gas given the moles, the temperature, and the pressure. To do that, we use the following formula:
[tex]PV=nRT[/tex]Where:
[tex]\begin{gathered} P=pressure \\ V=\text{ volume} \\ n=\text{ number of moles} \\ R=\text{ universal gas constant} \\ T=\text{ temperature} \end{gathered}[/tex]Now, we substitute the values for the final state. Since the process is isothermally this means that the final temperature is the same as the initial temperature. we get:
[tex](1.8atm)V_f=(2mol)(8.31415\frac{J}{Kmol})(243K)[/tex]We need to convert the pressure from atmospheres to Pascals. To do that we use the following conversion factor:
[tex]1atm=101325Pa[/tex]Multiplying by the conversion factor we get:
[tex]1.8atm\times\frac{101325Pa}{1atm}=182385Pa[/tex]Now, we substitute the value in the formula:
[tex](182385Pa)V_f=(2mol)(8.3145\times\frac{J}{Kmol})(243K)[/tex]Solving the operations:
[tex](182385Pa)V_f=4040.847J[/tex]Now, we divide both sides by 182385Pa:
[tex]V_f=\frac{4040.847J}{182385Pa}[/tex]Solving the operations:
[tex]V_f=0.022m^3[/tex]Therefore, the final volume is 0.022 cubic meters.
Part B. We are asked to determine the work done. To do that we will use the formula for isothermic work:
[tex]W=nRT\ln(\frac{P_0}{P_f})[/tex]Where:
[tex]P_0,P_f=\text{ initial and final pressure}[/tex]Now, we plug in the values:
[tex]W=(2mol)(8.31451\frac{J}{Kmol})(243K)\ln(\frac{0.21atm}{1.8atm})[/tex]Now, we solve the operations:
[tex]W=-8681.51J[/tex]Therefore, the work done is -8681.51 Joule. To convert to kilojoules we divide by 1000:
[tex]W=-8681.51J\times\frac{1kJ}{1000J}=-8.68kJ[/tex]Part 3. In an isothermic process the change in internal energy is zero, therefore, according to the first law of thermodynamics we have:
[tex]Q-W=0[/tex]Therefore:
[tex]Q=W[/tex]Therefore, the amount of heat is equal to the amount of work. Therefore, the thermal energy transferred is:
[tex]Q=-8.68kJ[/tex]